Puzzle 1 from a free game seen in Toronto Chinatown

The correct move is 1. R7.6

It seems that Red can win after
1.R2+5? A5-6 2.P3.4 A4+5? 3.P4.5 K5+1 4.R2-1 K5-1 5.R2.6 R4-2
6.R7.6 yielding a textbook endgame which is easily won for Red.
However, Black has a surprising defense:
1.R2+5? A5-6 2.P3.4 C4.5!
Red has to capture this cannon, because of 3.K5.4?? R4.6 checkmate
After 3.P4.5 A4+5 (Answer1) we see an endgame 2 rooks vs. rook and complete
defense (i.e. rook, 2 elephants and 2 advisors) This is a theoretical
draw if Black manages to achieve the correct setup for his pieces,
that is, rook on the open file and defensive pieces in the center and 
on the baseline. At the moment Black has not reached this ideal
defensive setup and Red can try to hinder Black from reaching the
best position for his pieces. There are a couple of lines where
Black has to be very careful but I don't see a forced win for Red.

With 1.R7.6! Red can secure the win:
1. ... R4-2 (other moves can't help because two rooks and one pawn
always win against one rook and complete defense)
2.P3.4 A5-6 (only defense against the threat 3.R2+5 A5-6 4.R2.4 checkmate)
3.P4+1 K5.6 (or 3. ... K5+1 R2+4 checkmate)
4.R2+5 K6+1 5.R2-1 K6-1 6.R2.6 (Answer2) with a simple technical win
for Red.

Puzzle 2 from a game Offinger - Burmeister played at the
Halbkugelnpokal 2003 in Magdeburg

The correct move is 1. C5.6

After 1.C5.6 R4.7 2.E3+1 Black has no good defense against
Red's plan 3.H6-8 followed by 4.H8+7, e.g.
A) 2. ... R7.1 3.H6-8 R1+3 4.K6-6 (Answer1)
B) 2. ... R7-3 3.C6.9 P3+1 4.C9.8 (Answer2) threatening e.g.
5.C8-5 followed by C8.6 or C8.4
C) 2. ... P3+1 3.P5.4 R7.9 4.H6+4 R9.4 5.R8-1 K4-1 
6.R8-2 P3+1 (or 6. ... K4+1 7.P7+1 threatening P7.6)
7.H4+6 R4-3 8.H6+8 K4.5 9.R8.6 (Answer3)

After 1.C5.9 R4.1 Black's chances are much better, e.g.:
A) 2.H6-8?? R1+3 3.E5-7 (or 3.A5-4? H6-4 4.K5+1 R1-1 checkmate)
3. ... R1.3 4.A5-6 H6-4 5.K5+1 H4-6 6.K5.4 R3-4 (Answer4)
B) 2.C9.8 R1.2 3.K5.6 E5+7 4.E3+1 C7.5 (Answer5) and Red is in trouble.

Puzzle 3 from a game Wang Jialiang - Meng Liguo

The correct move is 1. ... R2-6

Black's attack came to a standstill and Black has to look for his
own king. Red's rook, horse and king already have an active position
and Red is threatening to reinforce his attack by bringing his
cannon to the central file.

After 1. ... P5+1? 2.R6+4 Black cannot defend the central file with
E3+5 any more. Now the threat 3.C6.5 is acute, e.g.
1. ... P5+1? 2.R6+4 R2-6 3.C6.5 E7+5 4.C5+5 A5+4 5.R6-1 K5+1
6.R6+1 K5+1 7.R6-1 K5-1 8.R6-2 K5.6 9.H7+6 K6.5 10.R6.5 R2.5
11.R5+1 E3+5 12.R5+1 checkmate (Answer1)
There is no sufficient defense for Black:
A) 2. ... R2-3 3.C6.5 R2.5 4.H7-8 C1.7 5.H8+6 R5.4 6.A5+6 R4.5
7.H6+7 A5+4 8.R6-1 K5+1 9.R6-4 R5+1 10.H7-6 K5.4 11.H6+8 K4.5
12.R6+5 K5-1 13.E7-5 (Answer2)
B) 2. ... R2-3 3.C6.5 R2.5 4.H7-8 P9+1 5.H8+6 C1.3 6.E7-9 (Answer3)
C) 2. ... H9-7 (plans 3.C6.5 E7+5) 3.C6+4 A5+4 (or 3. ... R2-6 4.C6.5
or 3. ... R2-7 4.C6.5 R2.3 5.R6+1 checkmate) 4.R6+1 K5+1 5.R6-2 R2-7
6.C6.8 (Answer4)
D) 2. ... H9-7 3.C6+4 A5+4 4.R6+1 K5+1 5.R6-2 H7+5 6.C6.5 K5.6
7.R6-4 A6+5 8.C5+2 K6-1 9.R6+6 K6+1 10.R6-3 K6-1 11.C5-3 R2-8 12.H7+5 (Answer5)

After 1. ... R2-6 Black has good chances to repulse Red's attack, e.g.
1. ... R2-6 2.R6+4 H9-7 3.C6.5 E7+5 4.H7-6 R2-1 (Answer6)