LRM

To be honest I don't think you are going to get useful feedback. They will probably tell you they received over 500 applications many of which were excellent and that they had to make a difficult choice between similarly qualified applicants. I don't think they will disclose any actual selection mechanism such as GPA etc.

I don't think EU schools abide by the April 15 resolution in general.

I think you would have to be insane to turn down Yale for NW... Also the stipend at Yale is close to 60k from what I remember reading here or XJMR. As for Columbia grads going to industry I think it is in part due to the opportunity to do so by interning and networking in NYC, and the desire to stay in the city rather than moving to a college town in the middle of nowhere.

Chicago > Columbia > NW in my mind taking into account wider institutional prestige for industry jobs, which is where most people end up these days. Perhaps Columbia first if you really want to live in NYC.

The thing is people with competitive applications at Chicago are also likely to get offers from other top schools like Harvard, MIT, Stanford, Yale, Berkeley, Columbia, Penn, and others. I read somewhere that even MIT has to make twice as many offers as the intended class size, and other schools have to make at least 3 or 4 times as many offers. Below the top 10, that's probably even more.

This. Ultimately, for a given individual I don't think it hugely matters if he/she attended a top 10, top 20 or top 30 program. The median outcome even at top 10 schools is an industry job like Amazon these days... The biggest difference is basically "quality of life" during the PhD. Higher ranked departments offer bigger stipends, better facilities, and more research funding but not always better training and mentoring such as publishing with advisors.

I would choose Penn for the overall Ivy prestige and presumably better funding than a state school.

Unfortunately, visit days are not particularly informative. The structure of the program is very similar at most similarly ranked places, and presentations tend to sell the school more than anything. Probably the most useful part is informal conversations with current students. But the students that participate in the visit day are probably those with the most positive views overall. You will probably get a better sense of the culture of the department from anonymous posts on XJMR.

Not sure, I wrote undergraduate GPA as an example, you can probably compensate for it to some extent, but at the top 10-20 departments my guess is that anything below a 3.5 will instantly kill your application, simply because there are so many applicants with near 4.0 GPA, and a graduate degree, and a predoc, and a lot of math, and...

Tough one. If you are wealthy and/or would consider joining the workforce after the master's, then Chicago or Columbia for sure. The degree would pay for itself if you can get a high paying job from one of those places. For academia, I am not sure a master's will significantly improve your profile, unless perhaps you lack research experience and can work on a thesis and get good letters out of it. Your main concern is that if there are factors that hurt your application at better departments (such as low undergrad grades), it is entirely possible that an additional degree will not meaningfully address those and be a waste of time and money. But I wouldn't do a Ph.D. at WashU (or U Wash) or ND unless I was seeking emigration to the U.S. at any cost.

You can't go wrong with Harvard.

I would place Penn State above the rest, followed by Austin maybe. The only downside to PSU is that you will get annoyed when you hear "University of Pennsylvania."

Cornell > UT Austin, Ivy League prestige and all that + much higher stipend. Academic placement out of either will be questionable, so you might as well choose in function of prestige and industry opportunities.

No, I received an email from Graduate Admissions & Recruitment.

I don't understand what the point of these "unfunded first year" offer is. The vast majority of candidates will decline such an offer even if it means going to a lower ranked school. Best way for a school to end up with lemons it seems.