[SammyKid11]

Not to trifle with you on semantics...but on this flop, 77 is not the nuts. 86 is the nuts.

As for your line...
1) I'd fold preflop. You're seriously only going to play this the 12% of the time that you flop a set...you can't play 77 for overpair value against a raise, EVEN when your sevens ARE an overpair (which is rare). About the only other time you're going to call a flop bet without a set is when you flop an open-ended straight draw (cause a 7 really isn't even high enough to chase a 4-on-the-board flush). So what, let's be generous and say you'll stay to the turn 20% of the time with your 77. That means you need to be getting at least 4-to-1 on your preflop call. You're only getting about 3.28-to-1. Plus you're out of position against the raiser. SO...I think it's a losing proposition.

2) When you do flop your set, you look around and see a draw-rich board. You need to bet that. What you're hoping for is that the preflop raiser raises out SB...you call the raise and either push or check/push the turn. Instead what you've done is allow SB to call the initial flop bet with proper odds if he's got either a flush or an OESD (both are very decent possibilities). After he's called the initial bet, your raise isn't big enough for flush draws to properly fold. If you're gonna check-raise...with this stack, I'd make it check-push.

Since you didn't, I suppose your turn push is fine, though villain could easily have a straight and beat you here. But there's nothing else you can do...you can't check and give a free club to beat you, and I think you're really hard-pressed to bet/fold.