Well done! This is an example of a puzzle with the Pickaninny theme where a black pawn makes all four of its possible moves in the solution- 1 square forward, 2 squares forward, or capture to either side. In this case it was the black c-pawn in each of your four variations. When this occurs on the white side, it is called an Albino.

At last I have found how to get mate in 4 in all variations. I had difficulty with V-4.

V-1

1.Rb6! cxd6
2.d4 d5
3.Rd6 e5
4.Bxd5#

V-2

1.Rb6! c6
2.Rxb4 c5
3.Rc4 e5
4.Rxc5#

V-3

1.Rb6 c5
2.d3 c4
3.dxc4 e5
4.c5#

V-4

1.Rb6 cxb6
2.Bg1! b5
3.Bf2! Kxf6
4.Bd4#

1.Bg1 is incorrect.

V-1

1.Bg1! cxd6
2.Bd4 d5
3.Rd6 e5
4.Bxd5#

V-2

1.Bg1 c5
2.d3 c4
3.dxc4 e5
4.c5#

V-3

1.Bg1 c6
2.Rxc6 dxc6
3.d7 c5
4.d8=N#

V-4

1.Bg1 c6
2.Rxc6 dxc6
3.d7 Bxd7+
4.Kxd7 c5
5.Bxe6#

V-5

1.Bg1 c6
2.Ra2 c5
3.Rc2 c4
4.Rxc4 e5
5.Rc7#

I wanted to get mate in 4 in all variations but I couldn't do it.