Stat Assistant Professor

Yes, looking at the CV's of recently hired Assistant Professors at these schools is a good way to get some "baseline." It's not a clear-cut set of criteria for TT jobs, so you don't need [x] number of papers, exactly. It's more like if you have *at least* one paper in a top journal AND your research area is something that the department is interested in (so for example, a probabliitist with a very prolific record won't get an interview if the department isn't interested in hiring a probabilitist), then you will usually make it past the initial cut where they trim down all the applications into a set of 20 or so that they look at more carefully.  And the more papers you have in top journals, the fewer *total* number of papers you need (for example, an Assistant Professor at UPenn Wharton who joined the department in 2019 had "only" four papers, but three of them were in Annals of Statistics).
I don't think working with an Assistant Professor is necessarily an issue. There was one job candidate on the job market in the 2019-2020 hiring cycle who got like, 20 interviews, and her advisor was an Assistant Professor. She also got offers from UIUC, UNC, UFlorida, UMinnesota, Columbia, and probably others as well.

The research records of people at these places do not have to be anywhere near if you were getting a job at a top 50 statistics department. An average person at these places will have one article in a journal like Statistics in Medicine or Bioinformatics, and some more applied papers, approximately. I'd highly encourage you to take an hour or two and look at the top 50 or so LACs' faculty pages. Also pay attention to the years of the publications. Some I've seen don't have many publications before starting but had papers in progress.  I think liberal arts colleges have a lot of incentive to get people doing applied research in things like environmental science, health, social science, etc because they'll get students interested in research. Harder to get a college student to help you with your Annals paper.

Yes, looking at the CV's of recently hired Assistant Professors at these schools is a good way to get some "baseline." It's not a clear-cut set of criteria for TT jobs, so you don't need [x] number of papers, exactly. It's more like if you have *at least* one paper in a top journal AND your research area is something that the department is interested in (so for example, a probabliitist with a very prolific record won't get an interview if the department isn't interested in hiring a probabilitist), then you will usually make it past the initial cut where they trim down all the applications into a set of 20 or so that they look at more carefully.  And the more papers you have in top journals, the fewer *total* number of papers you need (for example, an Assistant Professor at UPenn Wharton who joined the department in 2019 had "only" four papers, but three of them were in Annals of Statistics).
I don't think working with an Assistant Professor is necessarily an issue. There was one job candidate on the job market in the 2019-2020 hiring cycle who got like, 20 interviews, and her advisor was an Assistant Professor. She also got offers from UIUC, UNC, UFlorida, UMinnesota, Columbia, and probably others as well.

Yes, looking at the CV's of recently hired Assistant Professors at these schools is a good way to get some "baseline." It's not a clear-cut set of criteria for TT jobs, so you don't need [x] number of papers, exactly. It's more like if you have *at least* one paper in a top journal AND your research area is something that the department is interested in (so for example, a probabliitist with a very prolific record won't get an interview if the department isn't interested in hiring a probabilitist), then you will usually make it past the initial cut where they trim down all the applications into a set of 20 or so that they look at more carefully.  And the more papers you have in top journals, the fewer *total* number of papers you need (for example, an Assistant Professor at UPenn Wharton who joined the department in 2019 had "only" four papers, but three of them were in Annals of Statistics).
I don't think working with an Assistant Professor is necessarily an issue. There was one job candidate on the job market in the 2019-2020 hiring cycle who got like, 20 interviews, and her advisor was an Assistant Professor. She also got offers from UIUC, UNC, UFlorida, UMinnesota, Columbia, and probably others as well.

For jobs at R1's (and R2's to a lesser extent -- though R2's do seem to care more about teaching and have a typical teaching load of 2-2), your publication record and your letters of recommendation are the most important aspect of your job application. Teaching doesn't matter as much, though you should put some thought and effort into the teaching philosophy. Most R1s ask you to submit a teaching statement, but it generally won't be given as much weight as the research. It's not necessary to have teaching experience to land an AP job at an R1. If you get a campus interview at an R1 or R2, they most likely won't make you do a teaching demonstration, whereas a PUI or a lectureship position would definitely make you do one.
Yes, it is possible to go straight from PhD to Assistant Professor, but your publication record would need to be especially strong in that case. One of my PhD classmates got an AP job at UMinnesota without a postdoc, but he had six papers by the time he graduated, including one in Annals of Statistics. If you have two or more papers in JASA, Annals, Biometrika, JRSS, Biometrics, etc. as a PhD student, then you could probably bypass the postdoc. Of course, the top journals count for more, so if you have two papers in Annals/JASA/JRSS/Biometrika, then you might be competitive with "only" three or four papers total.
I did not have any papers in the very top journals from my PhD, but got two either accepted or invited revision at JASA from my postdoc. So that helped make my profile a lot stronger.

I don't think it is necessary to involve undergrads in your PhD research. I know several faculty at PUIs and none of them did this for their PhD/postdoc. It's more that your application needs to convey that you understand that the main purpose of a PUI is centered around the undergraduate experience and education. So you should be able to come up with some project ideas that can involve undergrads.
Lecturer positions do not require any research at all (and hence, there are a lot of lecturers at math and stat departments who only have Masters degrees). A TT position at a four-year college will have at least some research/publishing requirements. But at a PUI, the teaching load would typically be anywhere from 2-2 to 4-4, so you aren't expected to do as much research (unless you're at a prestigious school like Amherst College).  They actually want you to spend most of your time on teaching and service to the department/college.
As for service, I would not expect most faculty applicants for AP jobs to have a lot of meaningful service to a university. But I do think PUIs would be very partial to applicants who have done some sort of mentoring (e.g. participating in a summer program where you taught a short course to students, programs where you mentored students from underrepresented groups in STEM, etc.).
Diversity is a plus for any type of faculty job, at both research universities and PUIs. I don't think industry experience matters all that much.  PhD institution doesn't matter as much as publications, letters of recommendation, and teaching experience, and in Statistics, that's true for either research universities or PUIs. The caveat to this, of course, is that people from more prestigious institutions are more likely to have more competitive profiles (especially for research). 

Lecturer positions at research universities are not tenure-track but are usually fairly secure. At my PhD institution, all the lecturers were able to renew their contracts. It seems like a sweet gig if your passion is teaching.
I would have to say that "teaching assistant professors" (i.e. that are tenure-track) are indeed extremely rare. If you want a TT job that prioritizes teaching, you should look for jobs at PUIs -- that is, liberal arts colleges and regional state schools that do not award doctorate degrees. It is a buyer's market though, so even these institutions will want to see some publications, and a lot of newly hired stats faculty will have done a postdoc. You would still need to publish as an Assistant Professor at a PUI,  but the publishing requirements would be considerably lower than at a research university for most of these schools (outside the very elite SLACs). Interdisciplinary articles also count towards tenure, and papers written with undergrads are especially well-received. The main criteria for tenure is teaching and service, though.
To be competitive for these jobs at PUIs, you should have some publications in respectable venues (not necessarily the top ones like Annals or JASA, but ones in places like Computational Statistics & Data Analysis, Scandinavian Journal of Statistics, etc.) and you should have taught at least one class as an instructor of record. Many PUIs ask you to submit teaching evaluations with your application, and they pay special attention to the cover letter and the teaching philosophy. So you really need to convey in your application materials your passion for teaching and how you can involve undergrads in your research. The search committees are always trying to weed out any applicants who aren't serious about the school's mission and who view the job as a "backup" or as a "stepping stone" to a job at a research university.
The campus interview for lecturer positions and AP positions at PUIs also always includes a teaching demonstration, so you would need to prepare for that (in addition to a research job talk if the job is tenure-track).
 

Lecturer positions at research universities are not tenure-track but are usually fairly secure. At my PhD institution, all the lecturers were able to renew their contracts. It seems like a sweet gig if your passion is teaching.
I would have to say that "teaching assistant professors" (i.e. that are tenure-track) are indeed extremely rare. If you want a TT job that prioritizes teaching, you should look for jobs at PUIs -- that is, liberal arts colleges and regional state schools that do not award doctorate degrees. It is a buyer's market though, so even these institutions will want to see some publications, and a lot of newly hired stats faculty will have done a postdoc. You would still need to publish as an Assistant Professor at a PUI,  but the publishing requirements would be considerably lower than at a research university for most of these schools (outside the very elite SLACs). Interdisciplinary articles also count towards tenure, and papers written with undergrads are especially well-received. The main criteria for tenure is teaching and service, though.
To be competitive for these jobs at PUIs, you should have some publications in respectable venues (not necessarily the top ones like Annals or JASA, but ones in places like Computational Statistics & Data Analysis, Scandinavian Journal of Statistics, etc.) and you should have taught at least one class as an instructor of record. Many PUIs ask you to submit teaching evaluations with your application, and they pay special attention to the cover letter and the teaching philosophy. So you really need to convey in your application materials your passion for teaching and how you can involve undergrads in your research. The search committees are always trying to weed out any applicants who aren't serious about the school's mission and who view the job as a "backup" or as a "stepping stone" to a job at a research university.
The campus interview for lecturer positions and AP positions at PUIs also always includes a teaching demonstration, so you would need to prepare for that (in addition to a research job talk if the job is tenure-track).
 

After you updated your post with your info, I would have to agree with the posters above that your profile is very strong. You could probably apply to all top 10 programs, and I'm sure you would get into at least a few of them. UC Berkeley and UW definitely seem plausible, as does Stanford if you can score well on the Math Subject GRE. If you are more flexible about your geographical preferences, you could probably get into really good schools on the east coast or midwest as well. 

- If you have not already taken it yet, take real analysis. And prioritize taking math classes rather than undergrad stat courses. Get A's in your math clases.
- Score well on the math section of the general GRE - ideally 164+. Unless you are planning to apply to Stanford or a school that "strongly recommends it," you don't need to worry about the Subject GRE.
- It seems like research experience is becoming more common for applicants to Stat PhD programs, so try to get some if you can.

Some math PhD programs like UCLA and UC Berkeley require incoming PhD students to take a "preliminary exam" or "basic exam" upon arrival that covers upper division undergrad material like proof-based linear algebra, real analysis (e.g.at the level of Rudin's "Principles of Mathematical Analysis"), complex analysis, abstract algebra, etc. You could spend some time practicing the publicly released prelim exams from different departmental webpages to make sure you will be able to pass these exams.
Other (usually mid-ranked or lower-ranked) math PhD programs basically re-teach you undergrad abstract algebra and analysis in the first year of the program before covering measure theoretic analysis and graduate-level abstract algebra and Galois theory. In that case, I think it is still worth it to review linear algebra, real analysis, complex analysis, and abstract algebra and practice the prelim exams from departmental webpages -- that way, you will be very well prepared for the first year of the PhD program, and that always makes the transition from undergrad to grad school a bit smoother.
This is what I did the summer before I began my PhD (though I'm in Statistics). Although I had previously taken Casella & Berger Mathematical Statistics, applied regression, etc. it had been almost two years since I finished a Masters, so I was a bit rusty. So I did every single past first year exam that was released on my department's website, and that helped a lot. I also reviewed real analysis and linear algebra before beginning.

I wouldn't worry about any gap less than five years. It's grad school, not a job interview, so nobody is going to get super worried about a year off - it's totally understandable given the circumstances.  The type of job you can get with a bachelor's degree won't be a big benefit to an application. If you could find a professor to work with from your undergrad institution to do some statistics stuff and get a better recommendation letter, you could try.  Since your grades and test scores are the main factors, unless you can do something to improve your recommendation letters with professors there isn't a whole lot to do.  I'd prioritize whatever job you can get to make some money which will make graduate school more enjoyable, or if money is not a concern, I'd do whatever you'd like to enjoy the next year the most. 

Some math PhD programs like UCLA and UC Berkeley require incoming PhD students to take a "preliminary exam" or "basic exam" upon arrival that covers upper division undergrad material like proof-based linear algebra, real analysis (e.g.at the level of Rudin's "Principles of Mathematical Analysis"), complex analysis, abstract algebra, etc. You could spend some time practicing the publicly released prelim exams from different departmental webpages to make sure you will be able to pass these exams.
Other (usually mid-ranked or lower-ranked) math PhD programs basically re-teach you undergrad abstract algebra and analysis in the first year of the program before covering measure theoretic analysis and graduate-level abstract algebra and Galois theory. In that case, I think it is still worth it to review linear algebra, real analysis, complex analysis, and abstract algebra and practice the prelim exams from departmental webpages -- that way, you will be very well prepared for the first year of the PhD program, and that always makes the transition from undergrad to grad school a bit smoother.
This is what I did the summer before I began my PhD (though I'm in Statistics). Although I had previously taken Casella & Berger Mathematical Statistics, applied regression, etc. it had been almost two years since I finished a Masters, so I was a bit rusty. So I did every single past first year exam that was released on my department's website, and that helped a lot. I also reviewed real analysis and linear algebra before beginning.

Oh oops, I somehow missed that the OP is applying to Statistics PhD programs rather than math PhD programs. In that case, disregard what I had written earlier. In that case, I would just review some Calculus, linear algebra, and maybe upper division undergrad-level probability and statistics and undergrad-level real analysis (these latter topics aren't as essential as having finesse with Calculus or LA, but basic familiarity with them might make the Casella & Berger mathematical statistics sequence a bit easier).

I think you can likely get into a Masters program provided that you have good letters of recommendation and explain your undergrad academic performance/emphasize any upward trend. Although your GPA isn't great, it is above the minimum threshold not to be auto-rejected. Masters admissions are also usually more lenient, since they are rarely funded. Finally, UT Austin is a well-respected school, so you have that going in your favor. 
I have also seen some individuals use Masters programs as "stepping stones" to get into PhD programs. These folks didn't have the best undergrad grades, but they did the Masters to show that they could get A's. Then they went on to do a PhD program (in math/applied math/stats). If you wanted to do that, you probably could -- though you would  likely have to temper your expectations for PhD admissions, in terms of what tier of programs you could get admitted to... but mid/lower-ranked schools might take a chance on you.
If your goal is just to get a Masters and then work in industry, I could see you getting into some MS program.

I think you can likely get into a Masters program provided that you have good letters of recommendation and explain your undergrad academic performance/emphasize any upward trend. Although your GPA isn't great, it is above the minimum threshold not to be auto-rejected. Masters admissions are also usually more lenient, since they are rarely funded. Finally, UT Austin is a well-respected school, so you have that going in your favor. 
I have also seen some individuals use Masters programs as "stepping stones" to get into PhD programs. These folks didn't have the best undergrad grades, but they did the Masters to show that they could get A's. Then they went on to do a PhD program (in math/applied math/stats). If you wanted to do that, you probably could -- though you would  likely have to temper your expectations for PhD admissions, in terms of what tier of programs you could get admitted to... but mid/lower-ranked schools might take a chance on you.
If your goal is just to get a Masters and then work in industry, I could see you getting into some MS program.

The top schools on the USNWR list look about right to me (NYU Courant, UCLA, and MIT). I would suggest you look at more than only rankings. Make sure there are actually enough faculty who are: a) doing research in your area of interest, and b) who have solid placements for their advisees. Are they placing their former students into top postdocs? If the school is a top-tier one like NYU, UCLA, MIT, Princeton, etc., the answer to part (b) is assuredly "yes." But beyond that, I would make sure there's a big enough group of researchers there. Some programs might be stronger in some area of applied math like PDEs/fluid dynamics or numerical analysis/optimization than others.

If you have strong performance in real analysis and can get great letters of recommendation, I think you could have a good shot at Biostatistics or Statistics programs. Here is a newly hired Assistant Professor of Biostatistics at UCLA who majored in German and Classics at UC Berkeley as an undergrad but then switched to Statistics/Biostatistics after his BA: https://andrewjholbrook.github.io/
I also personally know people who majored in biology, psychology, economics, and journalism  as undergrads but switched to Biostatistics or Statistics later on. Often times, these folks had to take the math prerequisites (the Calculus sequence and linear algebra) and then got a Masters first before going on to earn their PhD. But in your case, I think you could probably just apply directly to PhD programs since you  will have taken all the math prerequisites as a non-degree seeking student by the time that you apply.

If you have strong performance in real analysis and can get great letters of recommendation, I think you could have a good shot at Biostatistics or Statistics programs. Here is a newly hired Assistant Professor of Biostatistics at UCLA who majored in German and Classics at UC Berkeley as an undergrad but then switched to Statistics/Biostatistics after his BA: https://andrewjholbrook.github.io/
I also personally know people who majored in biology, psychology, economics, and journalism  as undergrads but switched to Biostatistics or Statistics later on. Often times, these folks had to take the math prerequisites (the Calculus sequence and linear algebra) and then got a Masters first before going on to earn their PhD. But in your case, I think you could probably just apply directly to PhD programs since you  will have taken all the math prerequisites as a non-degree seeking student by the time that you apply.

If you might be interested in biostatistics, I think you could also get into really good biostat PhD programs. I could see you getting into somewhere like University of Pennsylvania Biostatistics and biostat programs ranked higher than that as well.

Are you domestic or international? If you are domestic, I think you could probably add a few more schools in the range of NCSU through University of Minnesota and one other "reach" school besides UW.  With your profile, you can probably get admitted to Rutgers, ASU, and UCSB, so I would recommend applying to more schools around the range of TAMU.  
Just a note though: NYU has only 7 PhD students total, and they appear to all be international, so it might be extremely difficult to get into NYU. 

Some people on this forum seem to know people who do things like this, but I'll just give my experience/knowledge with tech companies.
If you're doing deep learning or something more CS-related, there are probably many more such opportunities. The ML/Statistics research group at Microsoft has 9 people, and many of them are not statisticians: https://www.microsoft.com/en-us/research/theme/machine-learning-statistics/. Google to my knowledge does not have statistics researchers outside of their normal data scientists (who do things that you might consider research?).  Facebook research does seem to have a bigger core data science team, which is more research-oriented.  The biggest key to being hired is likely having done the right type of research (deep learning, causal inference, networks, something that they have a need for) and having impressive publications.
The compensation is going to blow away anything in academia. Entry level data scientists with PhDs start at these companies at $200k+ total compensation.  I imagine researchers make at least that.  It is my impression that there are not enough of these positions for it to be a definite career path.  There are hundreds of people with PhDs from top 20 statistics programs just working as regular data scientists at these companies.

I don't think it should be an issue to transfer if you've passed your qualifying exams. At my PhD program, I know there was one person who transferred to University of Wisconsin Statistics and another who transferred to UNC Biostatistics. However, they both completed their Masters before starting at their new institution. If you're going to do this, make sure you're getting a Masters out of your current institution. Professors are busy people, so unless they had a very strong investment in you staying, they probably aren't dwelling on it and it's probably not as "awkward" as you think. People Master out of PhD programs all the time. 
Also, I wouldn't recommend trying to transfer in your first year. That definitely looks suspicious. It is safer to do so in the second year if you're going to do it at all. If you've passed your qualifying exams and done reasonably well in your coursework, you can mention those things in your SOP (or have one of your letter writers mention that), so the new program knows that your decision to transfer is not due to academic difficulties but due to evolving research interests, etc. 

I don't think it should be an issue to transfer if you've passed your qualifying exams. At my PhD program, I know there was one person who transferred to University of Wisconsin Statistics and another who transferred to UNC Biostatistics. However, they both completed their Masters before starting at their new institution. If you're going to do this, make sure you're getting a Masters out of your current institution. Professors are busy people, so unless they had a very strong investment in you staying, they probably aren't dwelling on it and it's probably not as "awkward" as you think. People Master out of PhD programs all the time. 
Also, I wouldn't recommend trying to transfer in your first year. That definitely looks suspicious. It is safer to do so in the second year if you're going to do it at all. If you've passed your qualifying exams and done reasonably well in your coursework, you can mention those things in your SOP (or have one of your letter writers mention that), so the new program knows that your decision to transfer is not due to academic difficulties but due to evolving research interests, etc. 

Your profile is actually pretty good. With a 3.9 in a math-y major from UC Berkeley, you should be in really good shape if you take a bit more math. 
I would recommend taking a few more math courses like numerical analysis and optimization, instead of econometrics or more statistics courses. For Statistics PhD programs, mathematical preparation is far more important than having taken a bunch of undergraduate applied stat courses (in fact, a not insignificant number of first-year Statistics PhD students have never taken a probability/statistics course before enrolling).  
Admissions is very competitive at the top Statistics PhD programs, so it would be good to apply to a semi-wide range of schools. But if you do well in real analysis, I could see you getting into a school like TAMU, Purdue, or Penn State (and you could likely get into a higher ranked school as well).