1.Rh5 e1=Q
2.Qxe1 g3
3.Qe5#

Yes, the problem with the 1.Rh4 Nh5 line is that the rook blocks the path to h8 for the queen if black takes at f4 with the knight on his second move. It took me a long time to even realize the line that goes 1.Rh5 Nxh5 2.Qh1 Nf4 was a mate in 3- I was blind to Qh8# and thus blind to even 2.Qh1 even though I found that move early, it seemed no improvement at all.

1.Rh5 Nxh5
2.Qh1 g3
3.Nxe2#

Assuming there are no typos in your solutions, then what about 1.Rh4 Nh5 line?

V-1:

1. Rh4 Nxe6
2. Nxe2+ Kxd5
3. Qh1++

V-2:

1. Rh4 Nxe6
2. Nxe2+ Ke4
3. Qh1++

V-3:

1. Rh4 g3
2. Bd6+ f4
3. Rxf4++

V-4:

1. Rh4 e1=Q
2. Qxe1 Kxc3
3. Qa1++

V-5:

1. Rh4 e1=Q
2. Qxe1 moves
3. Qe5++

V-6:

1. Rh4 Ne8
2. Qh1 moves
3. Nb5+(or 3.Nxe2+)