Puzzle 1 from an endgame textbook

The correct move is 1. C4+4

After 1.C4+4 Red has an easy win:
a) 1.C4+4 H5-6 2.C5+3 H6+8 3.P3+1 K6-1 4.P3.2 K6+1
5.P2.3 K6+1 6.K5-1 (Answer1)
b) 1.C4+4 H5+7 2.C4.7 H7+6 3.K5.4 and two cannons and
a pawn win against the lonely horse, e.g.
3. ... K6.5 4.C5-2 K5+1 5.C7-9 H6+8 6.K4-1 K5.4
7.C7.5 H8-7 8.C+-3 K4-1 9.C-.4 K4+1 10.P3.4 K4-1
11.P4.5 H7-5 12.C5.6 K4-1 13.P5+1 (Answer2)

After 1.P3+1 Red can win Black's cannon but that's
not enough for the win:
1.P3+1 K6+1 2.C4.1 H5+3 3.C1+2 C6.5 4.C5.2 H3+5
5.K5.6 K6.5 6.P3.4 H5-7 7.C1.3 K5.6 8.P4.5 K6.5
9.P5+1 H7+6 and, surprisingly, this is a position
Black can hold with exact defence. Red has to occupy
the central file with his king, but Black will not
allow that. (Answer3)
     

Puzzle 2 from a Chinese master game 

The better move is 1.R7-2

After 1.R7-3 R8+3 2.H3-5 (2.R7.6 R8.7 looks drawish,
2.H3+5?? R8+2) Black can save the horse with 2. ... H4-6
3.K4+1 R8+1 4.K4-1 R8.5. Although Red has a pawn plus after
5.R7.9 R5.9 6.R9+6 R9-2 7.R9.5 Black has no problems after
7. ... R9.5 8.P5+1 P9+1 (Answer1). Actually, Red has to be 
careful since his elephants can't defend very well against the 
infiltration of Black's pawn. A draw is the most likely outcome,
therefore Red could also play 8.R5.1 R5-1 with a clear draw.

In the actual game Red played 1.R7-2 and won after 
1. ... R8.4 2.K4.5 R4+2 3.H3+4 R4.9 4.E5-3 R9.2 5.R7-1 R2.5
6.E7-5 R5-1 7.H4+3 R5.4 8.P3+1 P1+1 9.P3.4 R4+2 10.R7.6 R4+2
11.K5.6 (Answer2). I don't think that Black's defence was
perfect, probably he had some drawing chances, but it's 
obvious that the practical chances for Red are much better
than in the line 1.R7-3.
     

Puzzle 3 from a game LIU Dahua - YU Youhua

1. ... R2+1 is the better move

Black is in trouble, in particular the cannon has serious problems. 
Black's goal is to free the cannon. Unfortunately, that's not possible
in all lines and quite often Black has to give one of his rooks for
Red's horse. Even in this case Black has reasonable drawing chances
if he can eliminate Red's pawns. As long as he has both elephants and
both advisors he can even hope to draw with one rook against two
rooks in many positions (that means Black might even sacrifice his
cannon for Red's last pawn in some positions). 

1. ... R2+1 with these lines:
a) 2.P4+1? C8.5 3.H3+5 (3.R2-4?? R2+5 checkmate) R8-4 and Black is better.
(Answer1)
b) 2.P4.5 R8.7 3.E1+3 R2.5 with good drawing chances for Black. (Answer2)
c) 2.H3+4 R2.6 3.H4+3 R6-3 4.R1.4 R6.7 5.R2.3, again a draw is the most likely
outcome (Answer3)
d) 2.R1.4 C8-2 3.H3+4 R8.4 4.H4+3 K5.4 5.R4.7 (5.R2-1?? R2+5) C8.9
6.R2-2 C9+4 7.R7+3 A5+4 (Answer4) Black's cannon is free. Red's attack
is still dangerous but Black's drawing chances are intact.

In the actual game Black played 1. ... R2.7 and lost after
2.H3+4 R7.6 (2. ... C8-1 3.H4+3 R7-2 4.R2.3 C8.1 R3-2 C9-1 R3.9 would 
have been  more difficult for Red, but this is still a won position, 
Answer6) R7.6 3.P4+1 C8-2 4.R1+2 (Answer5)