If the black knight can't move because it would put himself in check can the white king take the queen? Or would putting the king in D1 or D3 be an illegal move?
No. You cannot legally put the king into check, even if the piece that targets the king is pinned. Think about it as though you could capture kings. So if you take the Queen, the knight takes the king. Yes, that puts their own king in check, but they've already won by taking your king.