The (essentially) last hand of a single-table satelite

A single-table satelite with a $20 buy-in. Winner gets a seat in another single-table tournament.

I've been doing well, finding a lot of high pocket pairs--Kings, Queens twice, Aces twice, Jacks--and taking down big pots, sometimes busting people. My table plays down to me and Villain. He has $515 in chips, I have $580. Blinds 10/20. Villain is small blind and completes. I find myself with 5[img]/images/graemlins/club.gif[/img]7[img]/images/graemlins/diamond.gif[/img], and check. The flop comes 3[img]/images/graemlins/heart.gif[/img]A[img]/images/graemlins/diamond.gif[/img]7[img]/images/graemlins/spade.gif[/img], giving me middle pair. Thinking it unlikely that this flop has helped Villain, I lead out with a pot-sized bet, which hopefully, I think, represents an Ace. Villain calls the bet. Pot now $120. The call doesn't seem that strong, so I imagine Villain probably has an Ace with a weak kicker, which seems like a reasonable holding based on how he has been playing. The turn comes 7[img]/images/graemlins/club.gif[/img], giving me trips. I try to cash in on the Ace I think I've represented earlier and bet half the pot ($60). Again Villain calls. The river is the 2[img]/images/graemlins/diamond.gif[/img], which I doubt helps either of us. Villain's two calls to this point seem consistent with him holding an ace. Nonetheless, feeling they must represent some strength, I cautiously bet only 1/3 of the pot--which is $240 at this point--so $80. Villain raises all-in. After thinking for a while, I call and he shows me 7[img]/images/graemlins/heart.gif[/img]J[img]/images/graemlins/spade.gif[/img], beating my kicker and crippling my stack. Villain won the satelite in the next two hands. Your advice on my play? Was the line I took appropriate? Were my reads (just based on the board) decent? How about calling the all-in? Could I have laid down trips and left myself short-stacked?