PROBLEM NO 1

 See diagram below.

White to play and mate soonest.

How many moves?  Give analysis.

I don't know the answer, but my guess would be to move the knight to b4...c6 due to the outpost and e7 mate next move or capture the knight with black's queen. Black's pawns and king can't move, so the queen needs to do something to prevent this with either always checking the king or have the queen captured for stalemate.

Nb4, Nc6, then Ne7 (for white) seems to be the quickest checkmate (depends on where the black queen goes).

I don't know 7 or 8. Ill go with this :

What about Qxa4+ to get the stalemate or capture the knight with a check, as well as white's queen not being able to checkmate?

well 2 if black lets him

Well done, Zugzeit, you have covered the main variations, but we must also meet LearnChess's objection, which means:  we must still cover this variation:    1. Nb4 Qa8  2. Qf5 Qa5 .  How does White proceed then?