The correct move is 1.P6+1 

After 1.P6+1 Black has no defense, see e.g.:
A) 1.P6+1 C6.4 2.K6.5 C4.1 3.P4+1 C1+1 4.P4.5 (4.K5.4 also wins)
4. ... E3-5 5.P6.5 K5.4 6.P3.4 C1.6 7.P5+1 K4-1
8.P5.4 K4+1 9.P4.5 K4+1 10.K5+1 (Answer1)
B) 1.P6+1 E5-3 2.P6+1 K5+1 3.P4.5 K5.6 4.K6.5 E-+1
5.P6.5 E1-3 6.P+.4 (not 6.P5.4?? C6.5) C6.4 7.P5.4 (Answer2)
C) 1.P6+1 K5-1 2.P6+1 E3-1 3.P3.4 C6+3 4.P6+1 (Answer3)
D) 1.P6+1 C6.1 2.P6+1 K5-1 3.P3.4 C1+1 4.P6.5 (Answer4)

However, 1.P4+1? C6.4! is insufficient:
a) 1.P4+1? C6.4! 2.K6.5 K5.4 3.P4.5 C4+3 4.P3.4 C4.5 draw
(Answer5)
b) 1.P4+1? C6.4! 2.K6.5 K5.4 3.P6.7 C4.5 4.K5.4 E5+7 (Answer6)
and Black's elephants can idle, Red can't make any progress.
c) 1.P4+1? C6.4! 2.K6.5 K5.4 3.P6.5 C4.5 and Black will win a pawn.
d) 1.P4+1? C6.4! 2.P6.7 (or P6.5) C6+1 draw.
     

The correct move is 1.E7+5

Red has a pawn plus and the more active pieces. Therefore,
he has excellent perspectives in the endgame. However, he
has to be patient.
After 1.H4+5? the horse will be caught in a trap:
1.H4+5? C8.5 2.E7+5 H7-6
Red can try some tricks and may be able to draw the game
but he has no winning chances any more, e.g.:
a) 3.C8-5 H+-7 4.C8+2 H7+6 5.C8-2 (Answer1) with a possible draw,
although Black might try to find a win.
b) 3.H5+6? H--4 4.C8.9 A4-5 (Answer2)
c) 3.H6+8? K5+1 4.H8+7 K5+1 5.C8-2 A+-5 (Answer3)

The more careful 1.E7+5 gives Red good chances. There are
many lines in such an endgame but there is not much doubt
that Red's position is very promising. The game continued with
1.E7+5 C8.6 2.H6-5 C6-2 3.P7+1 E5+3 4.H5+7 K5+1 5.H7+6 E3+5
6.C8-5 H6-4 7.C8.5 K5.4 8.C5.6 C6+2 9.H6+8 H4-3 10.H8-9 H3+5
11.C6-1 K4.5 12.C6.3 C6.1 13.H4+6 (Answer4) and Red won after 
10 more moves (Answer4)

The correct move is 1. ... R2.7

1. ... R2.6? threatening 2. ... R8.6 checkmate looks like
a winning move. It seems that Red has to play 2.R1-8 R8.9
3.R2-1 R6.7 with a very good endgame for Black. However,
Red can launch a surprising counter-attack:
1. ... R2.6? 2.C6+9! K5.4 (or 2. ... E5-3 3.C6-1 E3+1 
[3. ... A5-4 4.R2.5 A6+5 5.R1.5 K5.6 6.R++1 K6+1 7.R-+2 K6+1
R+.4 mate, Answer3] 4.R1-8 R8.9 5.C6.8 K5.4 6.R2.6 A5+4 
7.R6+1 K4.5 8.R6+1, Answer2) 3.R2.6 A5+4 4.R6+1 K4.5 5.K5.6 R8.6 
6.K6+1 R--5 7.R6+2 K5+1 8.A5-4 R6.9 9.R6-1 K5-1 10.R6.1 (Answer1) 
and Red wins.

1. ... R2.7! also contains a mate threat: 2. C9.8 R8.6
3.K5.4 R7+3 4.K4+1 H8+7 5.K4+1 R7-2 6.K4-1 R7+1 7.K4-1 R7+1
8.K4+1 (or 8.K4-1 R7+1 mate) R7.6 mate (Answer4)
Again, 2.R1-8 R8.9 3.R2-1 yields a very good endgame for
Black. In contrast to the lines after 1. ... R2.6? Red
cannot win with 2.C6+9:
1. ... R2.7! 2.C6+9 K5.4 3.R2.6 K4.5 4.K5.6 R7.4 5.R6-3 R8-3
6.R1-8 R8.4 (Answer5) and Black will win the endgame.