A "3" would feel nice.
I think you allow the villian to keep the lead in the hand. Check call until river. Then you have a decision to trap or lead that street. Clearly if he checks back on turn you have to lead river with most run outs.
Yes, but if he checks on the turn you've lost an opportunity to build the pot. IMo, the only way most good players (villians) are going to keep betting is if they`re bluffing or have locked up two pair +.
It's interesting, if you ask 100 poker players how to play this hand, you'll probably get about 102 different approaches.