I find it unlikely, this would quickly earn them the reputation of being a turncoat. A bad reputation is pretty hard to shake off. Ten good deeds weigh the same as one bad deed.

Idk, it might be pretty hard to work at an institution where Facebook provides a lot of funding and not have that filter in to your mind at all, consciously or subconsciously. Let's say your considering a range of topics to pursue next and you know Facebook is a big funder, you may just choose not to pursue "is depression strongly positively correlated with Instagram use in teens" and pursue something else

This is exactly how manipulation is done and the reason for companies to do the funding.

It’s much less sinister than that. Facebook funds PhDs in some fields because they want experts in that field to a) exist, b) like Facebook.

Even the way you're phrasing it, sounds like PhDs don't get to a) exist without Facebook's blessing

It is generally the case that things don't exist without funding, correct. Different countries have different political consensus on how much tax money to pour into funding PhDs for basic research, but /all/ do to some extent as far as I know.

I think you should read Sapiens by Harrari, he writes about the interaction between universities, capitalism and democracy. Very illuminating to me.