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&amp;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.