What kind of reg are you against..

Against a straight forward nittty reg I would bet 1 time big ~ 7$ and give up if not improve.

Against a loose aggressive reg maybe you can try check/raise flop. Probably he will call the raise with medium pairs and 89s, 9Ts with backdoor flush. Turn will be check/check most of the time and you will get a free river .

This is a spot I get confused myself.

I think a bet and check-call are both fine lines to take, I'd most likely go ahead and bet instead of c-c and be like "oh we gotta fold now" on a TJQK turn. If the turn comes a high card I'd prob go ahead and turn my hand into a bluff and fire again and shove river because I go crazy like that sometimes .

The question you ask is a little too general as there can be so many variables - villains tendancies/stats, history, Turn and River obviously.

But as played with no info I think betting flop is totally fine we can get value from 33-55 straight draws and floats with 2 overs, 1/2 pot and re asses the turn, getting a fold on the flop is not a bad result either

If you have a stabby villain obviously the c/c becomes more attractive especially with the A kicker which he would barrel 100% if an A happened to appear.

check call flop is my prefered line here. This is not a hand that can take 3 streets of value and we would like to keep all his Ax left in his range.
If we check we will give him a free card with his two overs but that only has 12% eq so not the end of the world.

Think this is perfect hand to put in our check range to strenghten that a little bit. Putting better hands here instead will mess with our value range when betting.

I always go with betting flop here,intending to barrel the turn on any 6 or broadway card. It's obviously not great but we do get some thin value from floats that whiff the turn this way, and avoid capping our range in villain's eyes.

Not sure about what run outs we can go with a three barrel line though. I would usually be falling back on the hope villain doesn't choose to bluff any missed straight draws or stuff like 44 33