2 - Gianni Donati
Black has the move: Last moves were -1. Qe1-d1 d3-d2 -2. Qa5-e1 e4xPd3 -3. Qa8-a5 f5xPe4 -4. a7-a8=Q g6xPf5 -5. a6-a7 a7xBb6 -6. Ba5-b6 h7xPg6 -7. Rb6-c6 Rc6-d6 and the whole position unlocks. Black's other captures are c7xPb6 and b6xQc5.
So mate in 1 with 1. .. Qc6# and not with 1. Ng6?
3 - Mario Velucchi
Before the castlings, counting odd/even moves results in:
White: King: even, Queen: even, Rooks: odd, Bishops: even, Knights: even, Pawns: even
Black: King: even, Queen: even, Rooks: odd, Bishops: even, Knights: odd, Pawns: even
So white did an odd number of moves, black did an even number of moves. So black has the move. h#2 with 1. ... Kh8 2. Ne5 Rg8 3. Nf7#
9 - Stanislav Vokal
(a) The white queen and rook, which got captured by the black a- and f-pawn, couldn't have gotten out until a piece was captured on b3 or g3. So the f-pawn was still on the f-line, and the rook from a8 was still in its corner when the capture took place. So the only piece which could've gotten captured there was the black rook originally from h8. So the capture sequence is: a2xRb3; a7xRb6 and fxQg, or a7xQb6/fxRg; h2xPg3. So white can still castle, while black can't.
(b) Now the pawn on b6 could've come from c7, so the black rook from a8 could have escaped the northwest corner to sacrifice itself on b3. The a-pawn promoted after that, and went to g3. So both white and black may castle.
12 - Andrey Frolkin
(a) 1. f4 h5 2. f5 Rh6 3. f6 ef6 4. c4 Ba3 5. c5 Ne7 6. c6 dc6 7. b4 Bf5 8. b5 Be4 9. b6 ab6 10. d4 Ra5 11. d5 Rd5
(b) 1. c4 h5 2. c5 Rh6 3. c6 dc6 4. b4 Bf5 5. b5 Be4 6. b6 ab6 7. f4 Ra5 8. f5 Rd5 9. f6 ef6 10. a4 Ba3 11. a5 Ne7
(c) 1. b4 h5 2. b5 Rh6 3. b6 ab6 4. f4 Ra5 5. f5 Rd5 6. f6 ef6 7. c4 Ba3 8. c5 Ne7 9. c6 dc6 10. e3 Bf5 11. e4 Be4
Cyclic change (FCB,CBF,BFC) of pawn-captures.