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You say top 20-30, but your list is basically the 10 hardest programs to get into (except UC Davis, which I think is a reasonable target).  Your grades are obviously good, but I don't think there's a lot that makes your profile stand out.  Your math background and research are below average for people accepted to these schools. I'd classify all of these schools as big reaches, except maybe UCD which I would put in the "possible but not guaranteed" category.
If you don't want to go into a research-track job, then going to a program in the top 20 is not by any means required.  I think your best options are to either lower your expectations and apply to a much broader range of programs (there are many great programs all throughout the top 50 and even below the top 50!), or if you are not in a rush, get a master's and get some more background and research experience.

UChicago's MS statistics FAQ currently says they give 10% scholarships to most people - even 50% still leaves a huge cost for attending the program, which is why the person probably sounded disappointed.  Most people on this forum are applying to PhD programs and will get a master's degree for free, so any amount of money to pay would be disappointing when there are programs where you can go for free.

I think you'll get into many top 10 stats PhD programs and probably any biostats PhD program. I seriously wouldn't be worried about getting in to a good program if I were you.
Since you'll probably get in pretty much anywhere, try to focus more on program fit. Location, potential advisors, research areas, funding, curriculum, etc. are all important. Not all statistics PhD programs will be extremely theoretical, although most will require a similar degree of theoretical core coursework. If you like bio applications, a biostats program may be a good place for you, but many stats programs have heavy research in bio applications as well. Just depends on the program.

I think you'll get into many top 10 stats PhD programs and probably any biostats PhD program. I seriously wouldn't be worried about getting in to a good program if I were you.
Since you'll probably get in pretty much anywhere, try to focus more on program fit. Location, potential advisors, research areas, funding, curriculum, etc. are all important. Not all statistics PhD programs will be extremely theoretical, although most will require a similar degree of theoretical core coursework. If you like bio applications, a biostats program may be a good place for you, but many stats programs have heavy research in bio applications as well. Just depends on the program.

Not sure if you're trolling about your GPA.  Maybe you won't get into Stanford, but you'll get into plenty of top 20 programs and probably quite a few top 10.

Have you ever lived somewhere that's not California?  Seattle's weather concerns are overblown - it's less rainy/cloudy than many major cities on the east coast, and the winters are relatively mild.  It's not like going to Chicago or Minnesota.  But, obviously it's not Santa Barbara, and if weather is a huge concern for you, then only you know how much that would affect your happiness.
I think the best thing is to really think about your goals.  You should be able to get a decent job coming from any stats PhD program, but Washington is really in a different tier than the other two and I think you'd have more options, especially if you're interested in machine learning.  I think that holds even if you want to move back to California - you'l have more options in San Francisco after attending Washington due to the prestige of the program and your opportunity to do cutting-edge ML stuff.  On the other hand, I'm looking at UCSB's PhD placements and they are still pretty good: https://www.pstat.ucsb.edu/graduate/alumni
Location is huge though, and it's 5 or 6 years of your life, so if you think you'd be much happier in California, you can definitely be successful at any of the programs.

I think your research is fine - research is a big commitment, so I don't think it's worth it to go out of your way to get publications or anything like that.  If you have three letters from tenure-track professors in stats/math at one of the top schools in Canada, and they are very positive letters, you're in great shape.
A master's can definitely help, especially as an international student, but I don't think it's absolutely necessary.
If I were you, I'd just expand your list - apply to additional schools like UT-Austin, Ohio State, Illinois, that are great programs but not at the very top.  See how your results are, and re-evaluate if you need to.

You've basically listed all of the most competitive programs in the country.  You might have a chance at them, but this cannot be your entire list of schools.
Your profile is strong in that you have good grades from a good school and a a very strong math background - your research experience is sufficient, but without publications doesn't really stand out (at least from the details you've provided).  I'd widen your range through the top 30 on US News.

Start with US News, I'd focus mostly on programs in the 10-60 range.  See what locations you like and look at those department pages - most programs have a faculty page that will list research interests.  Most programs will have one or two faculty that focus on Bayesian stuff - programs that focus on Bayesian stats include Duke, Texas, Missouri, Iowa.  You'll have to dig a little harder depending on what exactly you mean by "machine learning."  Also look at biostatistics programs in the top 20, as your profile will play well there.
I'd talk to your professors, and focus on securing good letters of recommendation from research advisors - you go to a top 15 stats program, so try to take advantage of the vast connections of your department if possible.

I think your classification of schools as reaches/matches is basically correct.  Your profile is basically ideal, except for the lack of real analysis being a detriment at top stats departments.  I think you'll have a lot of success.

From their website, it seems like Math PhD students on the stats track would only have to take exams in Mathematical Statistics and Real Analysis, but do you really want to be taking a real analysis qualifying exam at a top 20 math department? Do you want to be wasting your time studying for PhD Math classes?
Personally, given my preferences, it would be a no-brainer to choose the data science program because I don't want to do that much math.  If you don't want to be a professor, I don't see many advantages unless you feel like you would get a lot of personal enjoyment and satisfaction from completing intense math classes.  I'd rather spend that time taking stats/data science electives and getting research experience, so I'd go for the data science PhD.
(There may be some areas/professions where the Math PhD will help you a lot in getting a position - for instance, maybe if you wanted to go into some type of quantitative finance position.  But for biostatistics or technology jobs, I don't think the difference between these programs would matter at all, and I think you could get a lot more research done/learn more practical stuff without the distraction of hard math classes that have nothing to do with statistics)

Since cost is not a factor, I think location is really the biggest consideration.  Maybe people who grew up in California have more associations with these schools, but you're not choosing between schools that are at vastly different levels of prestige like Stanford vs CSU East Bay.  I don't think the availability of slightly more electives is a reason to uproot your life if you don't really want to.  CSU East Bay would be my choice for this reason.

I would pick the undergrad thesis advisor and probably the lecturer you've had multiple courses with, assuming you have a good relationship with them.
Very few undergrads have done any meaningful statistical research, so I don't think an admissions committee would expect that of you. Your undergrad advisor will be able to attest to your general research ability and potential though, which is definitely important. It also sounds like you have a good relationship with them and have done some good work, so it's a no-brainer to me to pick your thesis advisor.
Between the lecturers, it probably doesn't matter much which one you pick as long as you've developed a relationship with them and they can attest to your good work and ability in the classroom.

I would pick the undergrad thesis advisor and probably the lecturer you've had multiple courses with, assuming you have a good relationship with them.
Very few undergrads have done any meaningful statistical research, so I don't think an admissions committee would expect that of you. Your undergrad advisor will be able to attest to your general research ability and potential though, which is definitely important. It also sounds like you have a good relationship with them and have done some good work, so it's a no-brainer to me to pick your thesis advisor.
Between the lecturers, it probably doesn't matter much which one you pick as long as you've developed a relationship with them and they can attest to your good work and ability in the classroom.

It probably doesn't matter too much.  The bigger problem people have is being too tied down to a specific area/professor and not being flexible.  If they require you to put some names in a certain box on the application, I'd just choose some people who look interesting and are still actively doing research that you'll be qualified to do (ie, don't put a probabilist if you haven't taken real analysis, etc)

Check the programs individually, but basically any program will let you master out, and many people transfer out after the master's and go to a new PhD program (this may be a headache and some people feel guilty about it, but it happens all the time).

Your profile is obviously great.  The only small criticism might be that your math background could always be deeper when you apply to a place like Stanford, but it's more than sufficient.
What's the reason behind only applying to UCLA/UCDavis's MS programs?   UCLA and UCD are great PhD programs (much better than UCSB).  I'd apply to Stanford, UW, Berkeley, UCLA, UCD, UCSD, UCI, UCSC and UCSB's PhD programs and I think you'd probably have some choices.

If you're at strong university known for grade deflation, I wouldn't be too concerned about the B's. A C in Real Analysis I might raise some eyebrows though, but I could be wrong. What are these "more difficult" classes you mentioned?
I hope ranking isn't your #1 factor when choosing which program to attend. Keep in mind you'll be in that department working with those people for 4-5+ years, so you need to make sure you also weigh other factors that affect your day-to-day life and happiness. On another note, I can't even fathom paying 50k/year for an undergraduate degree. There are a lot of solid programs outside of the top 5 though, so you definitely shouldn't feel bad if you end up finding a lower ranked program that is a much better fit in other aspects (people, research areas, location, funding, etc.)
It sounds like you're doing a lot this summer and in the fall that will greatly strengthen your application. In addition to those things, I would really focus on building solid relationships with your professors. Go to office hours, show genuine interest in their research, ask good questions. I didn't decide that I wanted to do a PhD until around the same time as you (around spring of my junior year), and I think one of the things that really strengthened my application even though I didn't have a super deep math background or any publications was that I had great relationships with some professors who wrote me very solid letters of recommendation.
If you want more specific feedback about which programs may fit your profile, I suggest posting some more details (like specific courses and grades, relevant previous research or other experience, research and location preferences, etc.)

I don't think you necessarily need a master's, no.  You have all the math and amazing grades/test scores from a good school.  I would definitely apply to PhD programs.

The two posters above are correct. A 4.0 from GPA from a top school UCLA and excellent grades in those math classes definitely make you qualified for a PhD program in Statistics. Research experience is a plus, but not having it won't necessarily hurt your application that much.
There is no need for you to get Masters, but if you have the bandwidth and the funds, you could potentially take a few additional upper division math/stat courses as a non-degree seeking student (for example, you took probability but did you take mathematical statistics?). You could take mathematical statistics, optimization, and another math class at a local university. This might further shore up your application as well, but it isn't strictly necessary.
Your letters of recommendation and your personal statement should emphasize your math ability and your grades in math classes. In addition, you might want to give some explanation for your motivation for wanting to get a PhD in Statistics after exiting law school. At least two of your letters of recommendation should be from math professors who can speak to your ability to succeed in a Statistics PhD program. I have sat on graduate admissions committees, and we really pay attention to math background and letters from professors who can speak to that. Good luck!

I don't think you can go wrong with the advice to "take more math", but I also don't think you can go wrong with the advice "make sure you don't do poorly in math classes."  Taking this class will, in all likelihood, not make or break your PhD admissions but getting a B or C will hurt.
Since you already have some analysis background, "doing analysis" as it relates to stats/ML is essentially using the ideas of convergence etc and applying them to probability/algorithms problems - you'll have to learn these things in the context of statistics anyways and you'd be better off just studying them in context.  For example, you'll take stats classes and learn things about convergence in probability -- you'd be better off learning about convergence in probability directly than taking a bunch of real analysis classes in "preparation" to do statistics stuff.  (of course, there are exceptions and if you want to do extremely theoretical research and go to UChicago or something, you probably want to take more math, but if that were the case, I think you probably wouldn't be asking this question)

One big reason I chose the program that I am currently attending is because the majority of PhD grads from my department go into industry and have great placements. My goal has pretty much been industry from the beginning, and the same is true for probably about half of my cohort. 
Biostatistics PhD programs tend to be more applied than Statistics programs, so as long as you have the basics (like calc 1-3, linear algebra, real analysis), you shouldn't worry too much about taking a lot of additional advanced math classes just to have them on your transcript. A lot of this is program dependent though, so just make sure you understand the level of mathematical rigor at each program you're looking at applying to by reading through the list of required courses. Keep in mind though that having a deeper math background can go a long way in putting yourself ahead of other applicants.

There is absolutely zero reason to think about your grade in this class unless it is going to interfere with you graduating.

90 hours? I'm not sure I believe that for one week, let alone a whole summer. In fields like stats and biostats, where virtually all of your work time involves intense thinking or doing (unlike lab-oriented fields where a decent chunk of "work" time is waiting for experiments to finish), it's just not possible for most humans to put in more than 50 or so productive hours per week. In grad school (at a top program), I was probably putting in about 40 hours/week spread across 7 days; some weeks less, others a bit more but rarely more than 50 and certainly never exceeding 60. I don't think I was way outside the norm.
If you have to put in anything near 90 hours per week on a regular basis to be successful in a program, then I would argue that you're in the wrong program (or, at the very least, should be looking for a different advisor).

I think it'd be worth it.  Competition is high these days and I think that's a pretty big signal of math ability, and also will look good in terms of showing you're taking going back to school seriously.  Will also help you personally during the program.  If you can afford it, I'd highly consider it.