Sorry, I don't know how to locate hand history or I would just post it. 52 ppl left 45 get itm. I start hand approx. 17 bb's. UTG min raises I have 77 in UTG+2. MP, HJ, and btn comes along for ride. Pot is now fairly equivalent to my stack size.

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UTG checks. I figure, if later positions had a pocket pair they would call a shove with here, they would have 3 bet preflop. I might get called with flush draw, so be it. UTG is checking because AK/AQ type hands don't look so swell and he doesn't have flush draw or he'd have to continue. I figure I am safe to shove this, possibly getting 8's T's maybe even J's to fold. My plan was to set mine but this flop deemed interesting.

Am I making the right shove here essentially picking up a double up if they fold out? What are villains calling range? FD, 9's, QQ+? (which we can rule out QQ+ from anyone who entered pot after me)

You have pocket sevens in the UTG+2, and your facing a UTG open minraise? I would just fold this preflop, you are vulnerable to get squeezed from someone, 77 isn't usually ahead of the UTG opening range, and your hand does not play great postflop.

On the flop, I would check because you shouldn't even be in this pot in the first place. You chose to shove the flop, not sure what you are hoping for, you fold out all worse hands, you are probably getting called by JJ+, since your shove looks very weak. A flush draw isnt going to call you, but thats actually a good thing since most flush draws have at least one over, but usually two.

Anyways fold preflop. As played, on flop you should check.

17 bb's is not nearly enough to set mine. It's annoying but you have to just fold pre.

As played the shove isn't that bad. You will pretty much only get called by better or flush draws, a decent percentage of the time you will however just pick up the pot.

But you shouldn't be in this position anyway. Vs an utg raise I would probably just fold the sevens (depends on the opponent, vs a very loose opponent you can shove preflop).

Calling to set mine is not going to be profitable with 17bbs, you need to be way deeper. Especially calling in this position is not a good move, as a lot of players are still left to act behind you and it's quite likely you are just going to be raised off your hand.

The being close to the bubble makes it an even easier fold preflop.

Fold pre.
Or push all-in based on opponent.

Count me in the "fold pre" camp. In general, if you're calling 10% or more of your stack preflop and you aren't trapping, you're making a mistake.