For the first case, it is clear that White cannot lose the game in any other manner than by flagging.
The USCF rule is as follows: In sudden death time control, a player on the move with two minutes or less of remaining time may stop the clock and ask the director to declare the game a draw on the grounds that the player has insufficient losing chances(§14H1).
But the key point here is that a claim must be made. If no claim is made, there is no adjudication possible, I would surmise.
The second situation is the same as the first IF NO CLAIM IS MADE, but unsure now is whether the claim can be upheld if it is in fact made. This is one of the more controversial USCF rule additions in the last 30 years.
See this article to help you decide.
Does black win or can/should a TD declare this a draw?
Clearly black cannot advance his position as it is actually *impossible* to do anything but slide the kings around. Yet assuming none of the draw conditions have been met, does the TD have a basis for declaring this a draw?
What about a situation like this one where it is technically possible for either side to horribly blunder themselves into letting an enemy pawn through though it cannot be forced.