Well Black's Queen knight must be the one on c5 as it was the pawn that initially occupied d7 and the Bishop that traversed it, meaning that the two moves for Black's Queen-side Knight must have been Na6, Nc5.
White's dark-squared Bishop was captured on b6, Black's Queen was captured on b3.
It looks like the Knight has to be on h3.
The black bishop is the only piece that traversed through d7 (the original pawn occupying it) . This means that the black queen would have to escape through a8 and the a-file. However since the black queens-knight would have to get to its final square by the a6-c5 maneuver, the black queen couldn't escape and be capture on b3 before the black knight is on c5.
This implies that the a1 rook can't be on c6, so that must be the h1 rook. For the h1 rook to escape, the knight must have moved to h3.
Yes -- that looks like it. Note that all of the d6 pawn push, the capture on b6 and the Knight's arrival on c5 have to have occurred before the Queen can get out to finally be captured on b3. This means that the caged rook must have also been there before the capture on b3 and so can't be the a1 rook.
Nice.
The White King's Knight has moved only once this game. Is it the knight on f3 or h3?
Know that a) The White king hasn't moved b) The black Queen's Knight has moved exactly twice, the square d7 has never been occupied or traversed more than once.