Is this a bad beat or did I completely misplay it?

[SonicReducer]

Last night I was plying in a three table NL Holdem tourney at a K of C Hall. 25 players to start, we were down to 16.

Blinds are 75/150
Stack sizes are approximate.

1. Button (1,200)
2. SB (1,500)
3. BB (1,700)
4. UTG (1,200)
5. UTG +1 (1,300)
6. MP (800)
7. MP2 (1,000)
8. Hero (785)

It's folded to me I call with Qh 10d. Button and SB fold, BB Checks.

Pot is $375

Flop comes 5s 10c Jh. BB checks I bet $200. BB raises me $435 (all in). I think about it, have a feeling he doesn't have a Jack and call it.

BB turns over 10h 3d.

Turn card is As. River is 3s.

Should I have raised pre-flop? If so, what other mistakes did I make? Should I have never played this hand to begin with?