Yes, this position can be won by black. Here's how:
These types of positions show the concept of controlling the million-dollar square. Now, what is that square? In this position, it is b6. Why is it called million-dollar square? It is called the million-dollar square because it's the most important square which decides whether black would win this or draw this position.
[FEN "6k1/8/KP6/8/8/8/8/6q1 w - - 0 1"] 1. b7 Qa1+ 2. Kb6 Qb2+ 3. Ka7 Qd4+ 4. Ka6 Qb4 5. Ka7 Qa5+ 6. Kb8 Kf7 7. Kc8 Qc5+ 8. Kb8 Ke7 9. Ka8 Qa5+ 10. Kb8 Ke8 11. Kc8 Qd8#)
How does Black win?
The basic idea to win these types of positions is to make the White king go in front of the pawn (on the queening square).
- After b7, the black queen starts giving checks.
- As soon as the white king is in front of the pawn(on the queening square), bring the black king closer to the pawn by one square.
- If the white king starts to run then give checks and force it to move in front of the pawn (on the queening square).
- Repeat steps 1,2 and 3 till you checkmate or win the pawn.
Learn more about Queen vs Pawn endgames
This is a standard Queen vs. Pawn endgame. The queen usually, but not always, wins. The standard idea is to check the enemy king repeatedly until it is forced to move in front of the pawn. This gives you a free move , which lets you advance your own king. If the opponent allows it, moving the queen in front of the pawn (...Qb8 in the diagram position) also wins.
It turns out that this general plan usually works, but fails if the pawn is a bishop's pawn or rook's pawn. In this case the defender can threaten a stalemate defense (e.g. if the pawn is on c2 and the king is on a1, then Qxc2 stalemates the king). There are exceptions and intricacies, though (see the article linked above).
In your case the pawn is a knight's pawn, so Black wins. If you know the general idea you should be able to find a win even against Stockfish, since it's hard to ruin your position. I would suggest trying it.
Yes, and its pretty simple. Black just plays Qg8 covering the queening square and white has no way to advance the pawn. Black will then advance his king until he captures the pawn and then it's a simple K,Q vs K endgame.
Most of the time that works in these types of endgames however if the pawn is a bishop pawn (c or f file). white can force a draw under certain circumstances ie if white can get to the 7th rank before black can move his king in. The reason is that there are stalemating opportunities. For example, white could move to the corner square (h8) and black can't take the pawn because it would be stalemate.