Statboy

The Casella Berger course is the core of what a statistician does (and what will be on your quals if your program has them), and most people don't use much measure theory at all, so definitely don't skip that. It's hard to give advice without knowing the specifics of your program, but there is probably no way to speed up the coursework. If you want to finish quickly, the best thing to do would be to be an RA and establish a good relationship with your advisor so you can get started as soon as possible on your dissertation. But you're not going to be able to do any statistics research without taking a Casella Berger-like class, so just do well in your courses.  Failing your quals by not focusing on coursework is one way to guarantee you won't graduate in four years. 
As for staying happy, I don't think putting an artificial timeline on yourself is going to help. This is going to be a slog, and it'll get done when it gets done. You can obviously try to go quickly (and this depends on your program - Duke is known for getting people out in 4 years) but I wouldn't want to be putting a strict timeline on this.
My one tip is to try to have some friends and activities outside of your program. It'll drive you crazy if you just think about statistics 24/7 - if you're able to keep some perspective on the grand scheme of life it'll help take some pressure off. 

You're about half way there. You still have to add lower schools - none of those is even close to a safety or even a safe target (Edit: I do think you're likely to be accepted to TAMU).  I'd take off Chicago and Columbia and add 5 schools that are below every school on your list. 

Your math background already looks quite strong for Biostatistics PhD programs and is already much more exhaustive than many Biostat PhD applicants -- the majority of Biostat PhD applicants have not written first author papers or taken graduate real analysis, stochastic processes, or algebraic topology. You would make a great applicant for Statistics PhD programs as well, if you are interested in applying to those.
Where did you complete your math degree? Is it at one of the top institutions in your home country (i.e. KAIST, Yonsei, KU, or Seoul National)? If so, I would advise you to apply to all the top Biostatistics PhD programs in the U.S. (and some Statistics programs too if you're interested in that). Admissions is very competitive for international applicants, but if you completed your degree at any of the top four universities in South Korea, you stand a very good chance.
 

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I was in your situation this past cycle, but more severe. The only stats course I took was 101 freshman year. The rest of my courses were pure math, like logic, algebra, and analysis, not even probability! I still got into some great programs. At visit days I learned that it's far more common to have a pure math background, although typically students have seen some level of mathematical statistics, so I do think I'm still in the minority, but I'm not quite as odd of an applicant as I initially thought. A professor at UW explained to me "we look for people that can learn statistics", as opposed to people that know statistics.
 
So, a background in pure math + graduate statistical theory class + research in probability and random processes actually makes you an ideal candidate from what I can tell.

If you are equally qualified as another similar applicant, the applicant with more statistics courses will be considered more favorably than you. There are plenty of statistics courses online. Is there any way you can strengthen your application?

I'm surprised about UNC as well. IMO if students are on your waitlist, you should at least make contact with them at some point prior to April. I can only see benefits to doing that, where are the drawbacks??

When I was a PhD student in a Statistics department, I collaborated with a few people in the Agricultural and Biological Engineering department and I knew another who did some work for the Nursing department. You will probably need to seek it out on your own, though, or volunteer to do it.
In my experience, the Graduate Coordinator or someone in another department will sometimes forward an email to the student listservs asking if any statistics grad students would be interested in doing statistical analysis, coding, plots, etc. This work will probably not count towards your PhD dissertation work but you will probably be listed as a third author on the paper that results from your work. A lot of Stat PhD students do not do this on their own volition though.

@Patrick McMahon 
First of all, I wanted to tell you that it takes a very strong person to be able to get through college when you have suicidal depression. My mother has had it on-and-off, and I know how it eats away at you every single day, and how it takes conscious effort to ignore how you feel/ think. You are doing a great job, and don't let anyone tell you differently.
Second of all, and to answer your questions, I believe you should be making academic decisions that will be beneficial to you in the long run (i.e., do not inundate yourself with a heavy workload if it would lead to you performing worse and worse). With your drive, and I can tell you have it, you will be able to succeed in a field like music (assuming you go on to grad school for it). However, if Math is truly your calling, you should focus solely on that and try to select only courses that are required for that major (and keep music as a hobby).
Regarding how grad schools look at people applying to a major when their undergrad was in a different major: It would be very difficult to do a Math PhD if you major in music for your undergrad. The courses required in a Math PhD assume you have a firm understanding of the Math fundamentals. Other students you would be competing against for the PhD position will most likely be students that majored in Math in undergrad (or a STEM major like Engineering, Physics, Computer Science, etc.). Thus, they would be preferable from the get-go because they would be less of a risk.
Tips for you in particular: 
-Try to talk to your counselling office about getting special accommodations for your test anxiety. Every university must abide by national laws regarding these issues.
-Consider doing your undergrad in 5 years instead of 4. I am doing mine in 5 years, and after going through the grad school application process, I can tell you that doing undergrad in 5 years will not be looked down upon by grad school admission committees. Doing undergrad in 5 years will give you the option to take fewer courses per semester/quarter and would give you more time to improve your mental health through extracurricular activities.

Got into UMass Amherst for the statistics PhD today, with guaranteed 5-year funding as well as first-year fellowship. That puts months of stress to an end, finally I'm officially going to grad school!

UF has several professors working on graphical models, including Kshitij Khare and George Michailidis. For job placements, I can tell you that this year, one of UF's fifth year PhD students got a tenure track Assistant Professor job at University of Minnesota and one got a job at Google out in California. And recent years have seen graduating students take postdocs at Columbia University, University of Pennsylvania, and Duke University, and industry jobs at Amazon, JP Morgan, Apple, and Siemens.
Feel free to message me if you have any questions specifically about UF. The rankings of OSU and UF are about the same, and either one can set you up nicely for a good academic position or industry position if you play your cards right.
 

I'm not sure how well-defined your research interests are, but besides vision research (an incredibly niche area of research), those aren't necessarily any more or less applied than other areas of statistics. There is very theoretical research in all those areas.  UF has a ton of faculty doing applied research, some of which they call machine learning, so I'm not sure why OSU seems to be a better research fit for you.  Your research interests might change as you learn more in your classes and go through a program too. Deep learning is a pretty niche topic in stats departments still, and most programs don't really have people doing research in it still.  All I'm saying is, before you base your decision on where to go solely on your perceived "research fit", make 100% sure you know enough to know those are really your interests.  Most students don't really know enough to know this when they start their programs, so I don't mean to sound condescending at all.
As for the other stuff, you'll probably be sick of structure after a few years when you are dying to be done with coursework. I know that for me personally, OSU having two separate years of important examinations was a negative for me.
Since you'll be spending most of your 20s in this place, I think that location is incredibly important, especially when the programs are so equal in reputation. If you like UF a lot more in that respect, I would strongly consider going there because it sounds like you would enjoy it more and still probably be able to do research there that you are interested in.  Keep in mind you'll be writing your dissertation with 1 or 2 professors so you don't necessarily need to love every professor's research. 
 

Resnick's book is pretty good, but it actually *is* a very mathematical, measure-theoretic probability book.  If you're preparing for the first year, this is probably overkill.  I'm guessing you would like something that will help prepare you to succeed in your first semester Casella & Berger class, especially if you have not had any probability course at all.  I'd take a look at Joe Blitzstein's Harvard Stat 110 class (https://projects.iq.harvard.edu/stat110).  There are lectures, homework assignments, tests, all available for free, and he is a genius at teaching the intuition behind everything, which you will really need.   It is an undergraduate course, but if you master the material, you will be set up for success in your Casella & Berger class.
As for calculus, I would just look up some integration by parts practice problems.  

If you want to be a financial engineer, go to NYU. If you want to be a statistician, go to Wisconsin. If you are totally undecided, go to JHU.

I can't fully answer your question, but I'm a current UNC biostat student, so I can offer a few clarifications about the program. First, (basic) measure theory is covered for one month, there is one test over the material (and measure theory is only half of the tested material), and then it's never brought up again (even on the qualifier). So I wouldn't go to UNC with the idea that you will get a lot of instruction in measure theory. I don't know anything about Duke, but for Duke stat to have less measure theory than UNC biostat, they would basically have to not cover it at all. 
Second, if you go to UNC, don't expect to do meaningful statistical research until after you pass your qualifying exam, which will be after 2 years if you come in without a relevant Master's (and if you pass on the first try). A small handful of students do a little bit of statistical research prior to passing their qual, but by and large, you basically won't do any work towards a thesis until passing. You will probably have to do some statistical programming for an applied project (i.e. GRA or training grant), but it's typically not the same as thesis work.
Third, this is more of a feeling than a hard and fast rule, but it seems to me that students don't really choose their faculty research advisor, the faculty choose their research students. So I wouldn't count on working with any specific research advisor at UNC. Also, for the higher profile faculty, they do tend to have a lot of students, and so many of them co-advise their research students along with a more junior faculty member with fewer time demands. 

It may be a good idea to spend some of your free time reviewing:
- basics from Calculus I-III (specifically: the derivative rules like product rule and chain rule, integration and rules for integrals like u-substitution and integration by parts; partial derivatives, double/triple integrals and changes of variables/Jacobian matrix, and sequences and series... you don't really need to review things like cross products or torque)
- linear algebra (both basic and proof-intensive)
- basic real analysis (at the level of Abbott's textbook).
It is natural for a lot of people to forget things from these classes if they have not encountered/consistently worked with those tools in awhile, but it is a good idea to refresh your memory before starting. I find the MIT OpernCourseWare (OCW) to have useful resources for reviewing these things. 

Got an offer from Iowa State Stat Phd today!, with TA.

Outside of a few super-star researchers who are constantly publishing in top journals and conference venues (e.g. Tony Cai, Michael Jordan, Jianqing Fan, etc.), most potential PhD advisors will probably not be publishing more than one article a year in JASA/Annals/Biometrika/JRSS-B (one every 1-3 years seems more common). It's probably important that they *have* published in these venues in the past, so they know what level of quality/novelty is expected for publication in these venues. But I wouldn't say it is essential to have a PhD advisor who has 1-2 publications in top journals *every* year.
To land a *really* good postdoc, I would say that nearing graduation, the PhD student should have either: a) 1 publication accepted/in revision in a top-tier journal like the ones you mentioned or a top-tier conference (ICML, NIPS, AISTATS), or b) 2 publications accepted/in revision in relatively good journals (say, top 10 like Statistica Sinica, Journal of Multivariate Analysis, etc.). And there should be at least one other work in preparation on top of that.
Any postdoc or PhD student who has 2 publications in JASA/Annals/Biometrika/JRSS/Biometrics will most likely be able to land an R1 job at a department in the USNWR top 80 (say).  Of course, if you want to get a job at Stanford, Berkeley, Harvard, Columbia, etc., you'll probably need more than that, with the most competitive job candidates having like, 4-5 in top venues. The people who tend to get the jobs at the most prestigious departments also tend to be graduates of similarly  ranked prestigious programs, so it seems difficult to "move up" significantly if your PhD is from a lower ranked institution (although a former PhD student of my advisor was able to get a faculty position at Duke, and our program is ranked a couple tiers below Duke -- in this case, the postdoc at a top 10 program most likely helped).
I myself am personally okay with going to a department in the top 80, but not necessarily the very top-tier, so I am working on getting two publications in very top venues at the moment.

I'm trying to assess potential research mentors at some of the schools I'm interested in, and particularly how competitive I would be for top postdocs (and eventually, fairly strong faculty jobs) if I excelled with them (I don't have a good benchmark for "excelled", but let's say it's at a level of some of their best 5 or 10 students [depending on professor age], but not necessarily at the level of their best student).  Let's say, for argument's sake, that I really wanted my first tenure-track job to be professor at a statistics or biostatistics department ranked in the USNews top 60 (combined).  To achieve this, would you want your advisor to have, say, 1 publication per year in JASA/JRSS-B/AAS/JRSS-C/Biometrika (similar level conference proceedings)?  How about 2?  3?  If h-index would even be useful (which I'm dubious of, particularly for people who consult on medical papers a lot) what would be a reasonable value over the past 5 years -- 15?  20?  If they are indeed a biostatistician and collaborate on a lot of medical papers, should that be more like 30, if any benchmark is useful?  Perhaps you have a favorite pet metric from the Publish or Perish software package, or else something better to look at (say, number of active grants/active NSF or NIH grants)?
How would these benchmarks increase if you wanted to teach at, say, a statistics department in the top couple tiers (e.g. Stanford through Duke, UPenn, and Columbia)?  A biostatistics department in the top couple tiers (e.g. Harvard/JHU/UDub through Berkeley and Minnesota)?  
If this question is just way too abstract and devoid of context to be able to answer, I'll note that I'm really asking this so I can figure out for myself how "deep" Duke statistics, UNC biostatistics, and Wisconsin statistics are in terms of their numbers of pretty strong and really strong advisors, respectively, so while I'll obviously entertain literally any response to this rambling monstrosity of a topic, I'd especially appreciate any speaking to that.

Nobody will rescind your offer unless you don't graduate. I promise.  I think you should keep the class and try not to fail so you still get the major, no question.  

I haven't heard back from Chicago or Columbia either. I checked Chicago's website (FAQ) and it explicitly says that they batch the applications and send out results (acceptances, rejections, waitlist) as soon as they review them, so if you haven't heard back it means they haven't reviewed your "batch" yet.
I mean, I want to believe them, but I can't help but think otherwise... 
And as for Columbia, I emailed the department today asking about my status. 
The struggle is REAL.

I haven't heard back from Chicago or Columbia either. I checked Chicago's website (FAQ) and it explicitly says that they batch the applications and send out results (acceptances, rejections, waitlist) as soon as they review them, so if you haven't heard back it means they haven't reviewed your "batch" yet.
I mean, I want to believe them, but I can't help but think otherwise... 
And as for Columbia, I emailed the department today asking about my status. 
The struggle is REAL.

haha. so true. spending all our time on thegradcafe doesn't help with maintaining a broader perspective...

. . .my goodness has it come to this?  Any applicant would be lucky to get into any of those schools.

Didn't get accepted by UChicago for an MS, which is totally fair. But not even being waitlisted despite having As in Real Analysis and Measure Theory, as well as some research experience? Also not shocking, but gotta say that stung a little.
Could it be because I only have one statistics class (Probability Theory)?
Blarg. Life goes on.