My thoughts on the first hand are that his raise on the flop is for information. He was trying to pick up the pot with a big bet hoping you would fold. Alot of heads up players are overly aggressive with just big cards. The turn was basically a continuation bet hoping you didn't hit anything to improve your hand, which he figured you would pitch into the muck. You push told him you were serious and he wasn't going to chase an over against a possible 2 pair.