Sometimes. My impression is that sometimes they're mostly funded by Yahoo instead, and other times perhaps Bing. But one way or the other they're always getting most of their money from one of the major search engines. So even when they have a contract with Yahoo, they have a financial interest in staying cozy with Google because when their contract with Yahoo expires they're going to take offers from both Yahoo and Google and, even if they ultimately decide to sign another contract with Yahoo, the offer they get from Yahoo will doubtlessly be competitive with whatever offer they receive from Google.
I believe there's currently no public information about that. They didn't get search engine payments from Google for several years, but with them replacing Yahoo with Google for US customers again it seems likely there's a new contract (although there were reports that the Yahoo contract had clauses that force Yahoo to continue paying them until 2019, so it's possible they just use that money right now)
The Mozilla Foundation is a US 501c3, all their finances are public record. I believe the most recent filed financial statement is for 2016 [0]. See also the summary document, the 2016 State of Mozilla Report, for higher level information [1].
