I'm not saying cbetting the flop is bad - but I'm tempted to consider that it might not be good. Do this:

List the entire hand range you would make the pre-flop bet with and for each hand in the range consider if it hits the flop. For each hand list what kind of hand it makes (set, overpair, underpair, TPNK, TPGK, TPTK, two par, flush draw, gutshot, OESD, combinations of the above). Consider how big a percentage of your hand range hits the flop. Basically do an ABCD range analysis of your range on the flop.

I suspect that your opening range in this position only quite rarely hits this flop. This would suggest that you need a relatively low ratio of bluffs to value bets. This brings me to the question: What would a bet with 77 be here? A bet for value? A bet as a bluff? A bet as protection? Do you want to build a big pot OOP with a hand that has bad reverse implied odds? I don't think a bet with 77 on the flop is for value, simply because I don't see many hands that 77 beats that would call you. It would be something of a bluff, and I'm not sure I agree with putting a bluff out when you cannot realistically represent having hit the board and when the opponent hasn't indicated to you yet whether he has. What you have on the flop is naked showdown value imo.

Because you're OOP and not hitting your set - and because I think the flop doesn't hit your range so you cannot realistically represent it - I think checking the flop is just fine. When the villain checks back he reveals weakness and in view of that the turn bet I think is ok. You are representing a random J (like AJ, KJ, QJ, JT, J9) that just hit and any weak hand that is not a J would be inclined to fold.

In the absence of the 7 on the river a flop plan of c/f, b/f, c/f seems sensible. Given the 7 on the river changing river action to b/f seems sensible. After the flop I don't want to put too much money in on a weak made pair.