GTO Brainteaser #6 -- Barreling and Bluffing

This weeks brainteaser is about optimal bluffing and bet sizing when barreling with a polarized range.  Due to work constraints I won't post the solution until around 7/5/2014.

There are 2 players on the turn, and the pot has 100 chips.  The Hero has a range that contains 50% nuts and 50% air and is out of position.  The Villain has a range that contains 100% medium strength hands that beat the Hero's air hands and lose to his nut hands.

For simplicity, assume that the river card will never improve either players hand.  You can also assume that the Hero is first to act (it turns out this doesn't actually matter).

If the Villain perfectly counters whatever betting strategy the Hero uses, which of the following two options is more profitable for the hero and what is the EV of each strategy?

1. Shove the turn (for 150 chips) with an optimal ratio of value bets and bluffs and check/give up with hands that we don't bet.
2. Sometimes bet 50 chips on the turn and sometimes barrel for 100 chips on the river, both with an optimal ratio of value bets and bluff.

Hint:  It is not +EV for the Villain to raise the turn against strategy 2 so long as the hero bets the nuts so you only need to consider strategies where the villain calls/folds with varying frequencies.

Bonus 1:  In general if the effect stacks at the start of the turn are X, what are the optimal turn/river bet-sizing and bluffing frequencies.

Bonus 2:  Suppose that 10% of the time the Hero's air improves to be the nuts and resolve the game.