I thought that if A1 goes up for a shot and B1 hits the ball with his hands and A1 loses possession in air and then regains possession in air, then he could come down. If he gets the shot blocked but never loses possession, I always thought it was a travel.

After the shot is taken, then blocked, of course he can come down with it. But your asking the same question, which has been answered. If the block prevents the release, jump ball-ap arrow. If the contact with the ball is not significant enough to prevent the release, travel. That being said you can error on the side of jump ball-ap arrow.