NO CATCH in any of these situations unless he first touches INBOUNDS.
Right, well not according to 4.3.3B:
*4.3.3 SITUATION B: A has third down and seven yards to gain at B's 30. A1 leaps near the sideline to attempt to catch a pass near B's 30-yard line. A1 is: (a) airborne trying to make the catch and is knocked backwards by B2 attempting to make the tackle and A1 lands outside the sideline at B's 32 or (b) airborne when he controls the ball attempting to complete the catch and is carried off the field by B2 landing out of bounds.
RULING: In (a), the pass is incomplete and the clock should start on the snap. In (b), the covering official must determine if forward progress was stopped in the field of play.
If the covering official determines that progress was stopped in the field of play, it is a catch and the clock should not stop. If stopped inadvertently by the covering official, the clock should be restarted on the ready for play. If progress was not determined to be stopped in the field of play, the pass is incomplete and the clock shall be stopped, to be restarted on the snap. (2-15-1, 2; 4-3-2)
I realize that (b) employs the dreaded word "carried," but the basis for the ruling seems to be the idea that progress was stopped, not that the player was carried off. And in general, the rule (no catch without something down inbounds) and the case (official's judgment whether progress was stopped) still conflict.