fujigala

A lot of business schools have professors doing statistics, yeah.  I know some pretty solid ML/stats people in business departments.  Psych, neuroscience, etc... you'll find people doing ML/stats.  Regardless, though, you have to find those people.  You absolutely cannot go into a program and expect that you'll be able to somehow get an advisor from another department.  I wouldn't go to a department that doesn't have at least a few people in the department itself that would be interesting to work with.
Part of this also depends on what you mean by "theoretical research."  What you consider theoretical might be more commonly thought of as methodology or even applied for some people.  

No, not really.   If you do a biostatistics PhD at a top 5 school, it may be possible to do some machine learning stuff, but it's certainly not going to be very theoretical.  What you should do is look at biostats PhD programs and go through every single professor's web page and see if they are doing anything that interests you.  Look at their papers in machine learning journals.  You will have to do the same thing for any stats programs you look at anyways, because ML research is still hard to find in most stats departments.

1. Yes.
3. If you think you're potentially interested in PhD programs later on (which it sounds like you are), the general recommendation is to apply to them directly. Most will consider you for MS if you get rejected, and worst case scenario, you can leave with a master's degree.

1. A 165Q is more than sufficient for MS programs
2. Yes--Ivy League schools are more competitive despite their ranking. But I think the small caveat here is that the more highly ranked programs have more employers recruiting (e.g., NCSU, UNC, and Duke will have a ton of employers recruiting from the research triangle, and because these programs are prestigious, the employers will look for stats people).
3. If you're interested in stat PhD, I recommend you post your complete profile to see if you could get into stat programs directly.

Well, depends... if you're a domestic student, then you might be able to get one of those jobs without a PhD -- and sometimes with only a Bachelor's. I have a friend who has BS in Biochemistry but he taught himself programming/hacking/etc., and with the "right" connections, he was able to enter the field of data science. Now he has been working in the field for quite some time, and managing data science/engineering teams. So if you manage to get your foot in the door and obtain the right experience, your degree may not even matter that much.
But if you're an international student, then it is *much* easier to get an industry job in the U.S. with a PhD. This is because it is easier to get an H1B visa with a doctorate rather than only a Masters.

As someone in industry, an MS is more than enough to get a job as a data scientist or biostatistician. If you're still hesitant about doing research then I highly recommend you go the MS route first then decide after working whether you want to pursue a PhD. 
I personally don't think a PhD is worth it for industry career growth. 
 
EDIT: 
I'd like to add that if you're looking for a PhD program than I'd look into theoretical MS Stats programs. If you're looking for industry then go to well known CS programs that have MS Data Science. Tech companies will be recruiting from there. From what my friends have experienced, the MS Data Science programs are a lot more applied than most MS stats programs. 

You do not have to commit to doing the PhD. You can enter the PhD program and leave with a master's. It's becoming more and more common.
Also, 4 years for a PhD (especially without a master's) is very fast. It's more like 5-6 years.

If you are contemplating getting a PhD in Statistics and your profile is competitive enough *without* the Masters, then I would recommend just applying directly to PhD programs.
But if you do insist on going the Masters route first, then the Masters in Statistics (or in Math/Applied Math where you can take the stats classes) would be the best preparation for a Statistics PhD program. For one, it might save time later as far as fulfilling coursework requirements -- you might be able to place out of all the first year classes. I have a MS in Applied Math but I took 4 statistics classes in my MS program, including both semesters of Casella & Berger and the applied statistics classes. As a result of this, I decided to try my PhD department's qualifying exam upon arrival (after spending maybe hundreds of hours practicing old qualifying exam questions), and I passed it so I was able to skip all the first year classes. That saved some time as far as degree completion. 
But even if you do repeat the first-year classes (applied stats and theoretical stats sequences) once you enter a PhD program, you will be completely prepared because you will have seen the material previously. 

Ranking MS programs into tiers is not a useful exercise. Yes, Chicago and Stanford have great programs, but they also have great PhD programs. The US News rankings are fine to follow and nobody is going to know the difference between Duke and Berkeley's MS program.  I think Stanford might be a reach but MS programs are not extremely competitive and I think you could probably get into a top 10 and definitely plenty of programs like UIUC.

AFAIK, Upenn does not have a terminal masters degree program in stat, and is earned on the way to a PhD, not necessarily a PhD in the stat department. That is unless you're talking about an MBA with a concentration in stat, which then has a different set of requirements different from an MS in stat. 

Master's admissions are much less competitive than PhD's since you pay for the degree. With your profile, if you can get your GRE Q to 166+, I think you should be competitive for the  top 10s.  I would definitely apply to top schools like Harvard, Berkeley, Chicago and Stanford as well as add some safer options in the top 20s.

You should know that master programs do not rank exactly like PhD programs (as in US news). I would like to use this opportunity to give a
TOP 10 Rankings for US Master Programs:
====================================
Tier 1:
1. University of Chicago (MA in Statistics)
1. Stanford (MS Statistics or Data Science)
---------------------------------------------------------------------
Tier 2:
3. Stanford (ICME, ML/AI/Data Science track)
3. Princeton (OFRE, MSE, MFIN)
3. MIT (Masters at OR or EECS)
---------------------------------------------------------------------
Tier 3:
6. UW
6. Duke
6. Upenn
----------------------------------------------------------------------
Tier 4:
9. Harvard
10. Berkeley 
====================================
 
This list is based on personal opinion but is generally influenced by popular opinion. For example, Stanford and UChicago is considered best 2 programs (personally I think UChicago is better than Stanford with thesis option, scholarship and ~50% admission rate to its PhD program). The other overlooked program at Stanford is the data science offered under its ICME institute as opposed to the one offered at stats department, which is also very good--there is an option to just switch to PhD at ICME upon satisfying some basic requirement. The next in line must be the MFIN or MSE at Princeton ORFE--at first glance it is untraditional for statistics but there is a good chance to switch to PhD program under ORFE which gives you excellent chance to study under top probabilists and statisticians. If you orientation is industry then ORFE gives you the best platform. The other excellent option to study statistics at master level is at MIT: either through EECS or OR's master programs. There are a lot of statistics/probability going around at MIT and lots of classes even though it does not have a dedicated statistics department. Next, UW, UPenn, Duke all have very good, very solid master programs in statistics.  The last category encompasses two better known schools. They are both good but curriculum is not as rigorous.  For example, Berkeley only has only 8 months, no thesis, and classes are really watered down stuff compared to PhD. Harvard is autonomous --more like a self-tailored sort of program--so it is harder to say.  
On your profile: 
Your course work and research are really good. I think you have a shot at Top 10 (maybe outside of tier 1).