So Greedo states the first time you attack and defend each round the first damage card is face up. Would that mean on TLT if you didn't get a hit the first time, but did on the second time, Greedo would not take effect? I suppose this would also apply to Gunner + Greedo on a YV-666?

Most definitely won't work on a Gunner attack.

Leaving aside the constant cluster-F that the double attack cards are, they pretty much come down as being two independent attacks, so the same would apply there - Greedo will only trigger on the first of the two attacks.