You can use chess engines
No. Black's pawns are just too far advanced.
Indeed it is a draw !
My first move is 1.Kb1
Actually, white could potentially win if he were first to move:
1. Re6 Kg7
2. Rxe5 Kf6
3. Rf5+ Kg6
From there, white could start stripping pawns from black without leave.
In fact, it doens't matter who moves first. White will win.
No, it's lost for white, black has the plan to move the king to g7 or g6, rook moving to h8 to h2 to xg2, there's no stopping the black g-pawn from queening because the white rook and white king are both too far away. White can munch pawns e5 and a5, but it's resignable.
But you might want to check with an engine, they generally find some fantastic resources in lost positions.
Kb1 Kg7 (I am using a engine)
1.Kb1 Kg7 2.Rh6
It's not drawing. Not all closed positions are draws!
1.Kb1 Kg7 2.Rh6 Kxh6 3.Kc1
1.Kb1 Kg7 2.Rh6 Kxh6 3.Kc1 Kg6 4.Kd1
1.Kb1 Kg7 2.Rh6 Kxh6 3.Kc1 Kg6 4.Kd1 Rh8 5.Ke1
1.Kb1 Kg7 2.Rh6 Kxh6 3.Kc1 Kg6 4.Kd1 Rh8 5.Ke1 Rh1+ 6.Ke2
1.Kb1 Kg7 2.Rh6 Kxh6 3.Kc1 Kg6 4.Kd1 Rh8 5.Ke1 Rh1+ 6.Ke2 Kh7 7.Ke1