Call the first hand without doubt - he is short-stacked so he may be being more cavalier than if he'd had $30 to bet. I don't think he's got a better hand than you.

Second one, you fold. He wasn't scared by the first King - indeed he called a decent sized bet, so he obviously felt that even if you had a king he was in a strong position. He didn't have KK as he would have raised you, so perhaps he had AK (unlikely as you had AA) or KQ, or perhaps Kxs. Regardless, your check on the river gave him permission to push, which he duly did. I don't think there was a lot more you could have done.