My thoughts:

Villain probably has: 77+ A2s+ ATo+ KTs+ KJo+ QJs+

Pre-flop: I call because if I 3bet I will only fold out weaker hands, all the stronger cards he will continue with beat me. By calling I can get value of KJ-, QJ-, and good cards for second nutflush and possible straight where I can extract value from because he holds a lot of AT+ 

Flop: Fairly certain we both missed, he either A) c-bets for value here with 77+ or B) c-bets as a bluff. I call because if I hit my K or Q on the turn he will probably fire another barrel with his pair, which I beat most off or he checks to me and I can float him.

Turn: My equity increased (or not... see extra question) because I now have a flushdraw, villain c-bets the turn again with probably his 77+, A3 or with his flushdraw. (Extra question: If the river gives another diamond and he goes all-in he will probably have Adxd right, because I block all lower flushes?)

River: Good card for me, he missed his draws and I might get value of lower pairs than mine. He checks to me and I bet, he then goes all-in. What can he have? AA, AK, KK, A3?