Geococcyx

You can definitely find the sorts of statistics you like in biostat departments.  Just to drive the point home, it looks like you did some research on Bayesian classification -- a reasonable example of how that may work in biostat may be Briana Stephenson, who worked on Bayesian clustering methods at UNC biostat (and now is a prof at Harvard biostat).  Out of your other possible interests from your profile: plenty of biostat folks do causal inference (e.g. Hudgens, also at UNC), and for your more machine-learning sorts of approaches, one example may be the sort of precision medicine work that Kosorok at UNC biostat and Laber at NC State (which is combined stat/biostat) have done.  These examples are obviously north carolina-slanted, but this isn't a unique situation (e.g. Witten does biostat ML at UW).  To StatsGOd's point, most of these applications don't require that you be an expert in biology (and correspondingly, there are plenty of geostatisticians who aren't earth scientists and whatnot).
I think bayessays has some nuanced opinions about whether biostat departments are the right choice for you in terms of being miserable or not; I'll let them weigh in if they desire and eschew addressing it myself.

They don't really teach any biology in biostatistics programs. It's only necessary to learn/re-learn bio if you're interested in statistical genetics.

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Yes, that Stats 300C class at Stanford is one possibility. I would say that a PhD-level advanced inference class should focus less on topics like UMVUE, Neyman-Pearson Lemma, admissibility, etc., but more on stuff like theory for shrinkage methods, convex/nonconvex optimization, reproducing kernel Hilbert spaces, resampling methods, etc. That's because the latter topics are more of current interest and are active areas of research.

I think you could get into some of the big state schools ranked between 20-50, which are great programs.  I definitely don't think it's worth doing a master's degree and spending all that money.  But I think it's probably unlikely to get into most the programs you've listed (especially Chicago/Berkeley/Penn) because of your combination of lacking advanced math and coming from a non-top school.

Are you sure there is no "advanced calculus" class at your school? It's hard to imagine a top 150 school that does not offer a class where people learn to prove basic things about calculus. 

The classic book for linear models in a PhD program is Christensen's Plane Answers to Complex Questions. 

I haven't seen anyone mention the financial aspect. When I applied, on average, each PhD application cost about $90. You should really think about if you can/should/want to spend over 1 grand on applications (on top of GRE fees). And be honest with yourself about your application package; my recommenders were maybe a little too nice to me. (Personally, I regret wasting a couple hundred dollars on applying to Top 5 programs.) Maybe you can better allocate that time/money elsewhere, eg. retaking a standardized test, networking, building up your academic/research/work experience.

I won't comment on your application's strength, but some other folks (and their respective schools) where you might find spatiotemporal modeling: Daniel Simpson and Vianey Leos-Barajas at Toronto (I know someone already recommended Dr. Simpson, just included for completeness' sake), Rick Schoenberg and Karen McKinnon in the UCLA statistics department (not sure how easy it is to do cross-advising between biostat and stat there), Brian Reich at NC State, and Matthias Katzfuss at Texas A&M. Additionally, for wearable technology, I tend to think of Vadim Zipunnikov at Johns Hopkins and maybe Julian Wolfson at Minnesota, but there are probably many folks I just haven't found.  
This isn't really my area, so I hope these are good enough folks to look at as a start until you can get into the papers and see who's getting cited, and that I haven't accidentally misrepresented anyone's research interests. Most of them do at least focus on environmental applications, which tends to be a good start.

The last four schools you listed are the ones I definitely think of as having spatial/environmental focuses.  I'd also look at Texas A&M.  Your math background is less than ideal, with your only proof-based grades being the C and B in real analysis, but as a domestic student from a top 10 program, I still think programs like Mizzou and Oregon State are likely safe options, and you have a decent shot at a lot of the other PhD programs.

I feel like UCLA stats would definitely want to see a formal real analysis course. Maybe not as much with UCLA biostats. I would maybe suggested you reach out to the grad coordinators of the programs that you're interested in and ask their opinion.

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.

Stat Assistant Prof answered some of your non-teaching questions here: https://forum.thegradcafe.com/topic/116685-good-productivity-benchmarks-for-a-strong-research-advisor 

Some of OP's questions seem like they might be answered with the slides and video recordings from eCOTS's workshop on preparing for teaching-focused faculty positions; here is the website to access that: https://preparingtoteach.org/agenda/ (there was recently a review of this posted via Sara Stoudt and Mine Cetinkaya-Rundel, posted here for a quick summary: hathttp://www.citizen-statistician.org/2020/06/preparing-to-teach-2020-what-did-we-learn/). That said, they seem to largely agree with our experienced posters, so this may not be worth your time.

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.

Besides Stanford, these schools generally will admit qualified domestic applicants without a subject test. 

Some of OP's questions seem like they might be answered with the slides and video recordings from eCOTS's workshop on preparing for teaching-focused faculty positions; here is the website to access that: https://preparingtoteach.org/agenda/ (there was recently a review of this posted via Sara Stoudt and Mine Cetinkaya-Rundel, posted here for a quick summary: hathttp://www.citizen-statistician.org/2020/06/preparing-to-teach-2020-what-did-we-learn/). That said, they seem to largely agree with our experienced posters, so this may not be worth your time.

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.

You should apply anywhere you want, I'm sure you'll get into some top ten PhD programs.