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If I had to guess I would say Q9 here.

QQ, 99, 77 hed raise PF. 34% VPIP I dont see Q7s or 97s.

The problem with calling the flop and raising the turn is what he thinks you have.

for the set: 99, 77, QQ
for the TP with draw: AdQ, AdA

If my read is right you have 5 ways for the set and 9 ways to have AdQ or AdA. If I was him id reraise with Q9 if you just called the flop. If you have like AdKd would,nt you cap the flop?

Now because you capped the flop can you raise the turn? I think you can if you intend to call down. You can only hit a K in this case because I think he has Q9. But I dont think he can reraise you on the turn, you capped that flop and with the diamond it adds more hands to the mix that can beat him. AdKd, Adjd, QQ, 99, 77, KdJd and so on.

check unimproved on the river or bet if you hit a K.

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Interesting hand. With a 22% PFR, I don't think he has a set, nor do I think he would CR it on the flop. I think we can rule out AQ, KJs, or any broadway PP. I put him on something like Td9d, Td8d, Q9 or Q9s, 97s, JTo, J8s in about that order. JdTd is a possibility (though I raise that in MP and I run slightly under his PFR) and I'd would play the flop like that with that holding. The flop CR and resulting cap lends me to think he has a big draw.

Smart money would fold the river. I don't see much in the way of possible holdings you beat after that turn, even at 10:1 odds. Given that I'd probably call down. JTo is the only thing you may be ahead of, and I doubt he caps the flop with it (though I've certainly seen it before).