U = bet unit, P = pot size in units

1. U/(P+U)
2. A,K,Q,J
3. 1/1.5 = 2/3, so villain has to call at least 1/3 == 4 cards
4.
EV betting: he calls 2/12 we beat, he calls 4/12 we lose, he folds 6/12
EV(bet) = 2/12(P+U) - 4/12(U) + 6/12(P) = 8P/12 - 2U/12

EV checking: we beat 8/12, lose to 4/12
EV(check) = P*8/12 - 0*4/12 = 8P/12

5. Will 8P/12 - 2U/12 ever be greater than 4P/12? No, unless U is negative (which doesn't make sense).

6.
EV betting: he calls 3/12 we beat, he calls 3/12 we lose, he folds 6/12
EV(bet) = 3/12(P+U) - 3/12(U) + 6/12(P) = 9P/12

EV checking: we beat 9/12, lose to 3/12
EV(check) = P*9/12 - 0*3/12 = 9P/12

7. The EV is the same, so it doesn't matter which he chooses from an EV perspective.

8a. Once your holding is behind the villain's calling range, you can't profitably bet anymore.
8b. As your hand gets worse against the opponent's entire range, you can't bet for value once you are behind their calling range. But you can bluff with the worst of your hands, depending on their calling frequency and your bluff size (because fold equity alone will make it more EV than checking).

This implies that we should bet a polarized range -- bet the top of our range for value, and the bottom for fold equity. The exact values depend on the villain's calling range and the bet size. After some Excel magic for a pot size of 2 units:

Bet: { A,K,8,7,6,5,4,3,2 }
Check: { Q,J,T,9 }