University rank/stature - How much does it affect one's career post-Ph.D?

Let me answer as a theoretical computer scientist with former PhD students in tenure-track academic positions and many years of experience on faculty hiring committees. (However, my understanding is that the selection process at industrial research labs like IBM T.J. Watson, Microsoft Research, Google Research, AT&T Research, etc., is really not that different from academic recruiting.) As always, take my advice with a grain of salt; I'm as guilty of confirmation bias as any other human being.

Nobody in theoretical computer science cares where you got your degree. Really. We. Do. Not. Care. We only care about the quality and visibility of your results. Publish strong papers and give brilliant talks at top conferences. Convince well-known active researchers to write letters raving about your work. Make a good product and get superstars to sell it for you. Do all that, and we'll definitely want to hire you, no matter where you got your degree. On the other hand, without a strong and visible research record, independent from your advisor, you are much less likely to get a good academic job, no matter where you got your degree.

(This is less true in more applied areas of CS, in my experience, mostly because it's significantly harder for PhD students in those areas to work independently from their advisors.)

But. Faculty candidates are necessarily judged by people who are not experts in their field. Without the expertise to judge whether your work is really good, those people must look at secondary data that correlate strongly with successful researchers. One of those secondary characteristics is "pedigree". Did you get your degree at MIT, Berkeley, Stanford, CMU, another top-10 department, or somewhere else? (What's an "Ivy League"?) How good/famous is your advisor? If they're really paying attention: Where did your advisor's other PhD students get jobs, and how well are they doing now?

Fortunately, most good departments do make a serious effort to understand the quality and impact of applicants' results, instead of relying only on secondary data. Also, secondary data matters considerably less once you actually have an interview.

And. In my experience, where you get your degree is strongly correlated with successful research. I got my Master's degree at UC Irvine in 1992 and my PhD at UC Berkeley in 1996. The biggest difference I saw between the two departments was the graduate-student research culture. Every theory student at Berkeley regularly produced good results and published them at top conferences. When the FOCS deadline rolled around each year, the question I heard in the hallways from other students was not "You know the deadline is coming up?" or "Are you submitting anything?" but "What are you submitting?", because "nothing" was the least likely answer. Everyone simply assumed that if you were there, you were ready and able to do publishable research. Publishing a paper wasn't exceptional, it was just what you did. That cloud of free-floating confidence/arrogance had a huge impact on my own development as a researcher. I've seen similar research cultures at a few other top CS departments, especially MIT, Stanford, and CMU. (Caveat: This is an incomplete list, and there are many departments that I've never visited.)

tl;dr: Yes, getting a PhD from a top department definitely helps, but more by helping you become a better researcher than by making you look better on paper.

The short answer is that it can matter fairly significantly in where you get your post-doctoral fellowship and eventual professorship, and it will matter very significantly if you choose to follow a career outside of academia.

When looking for a job in academia, potential employers will look at many factors, including publication record, research success, research track, who your advisor was, etc. The school is important but other factors are involved.

When looking for a job outside of academia, they will look at your GPA and the name of the university from which you graduated. In this case, your university could easily be a "make it or break it" part of the deal.

The larger the workplace, and the more applicants they're responsible for screening, the more important a role the academic pedigree will end up playing. A small business with a handful of applicants—or a professor hiring a single postdoc—probably doesn't need to screen out candidates as efficiently or as ruthlessly as someone that gets dozens or hundreds of applications for an opening.

To point out specific data points, my previous employer had a "preferred" list of schools for its technical hires; if you went to a school that wasn't on the list, it was a lot harder to get hired, and some hiring managers wouldn't even try to go through the work needed to get around this ruling. In some cases, this even applied to people who had been out of school for decades!

So, your pedigree is almost never a disadvantage; and as I have been told by many an academic, it can be of enormous benefit to you, particularly if you make the most of your opportunities at a big-name school.