Interesting if h5 before playing the Ke6-Kd5-Kc4 plan would be an improvement....everything depends on whether white has enough time to create a passed pawn.
Black wins here.
LESSON: ALWAYS IMPROVE YOUR POSITION TO THE UTMOST BEFORE MAKING THE FINAL BREAKTHROUGH.
If white goes to take the pawns on the queenside with Ke5 after Kxc3 and Kxe4, then after Kc4 Kd6 Kxd4 Kb6...I have not analyzed it. It may be drawn.
Start at the starting position but the critical idea is black to get his king to c4 and it seems winning - if played right.
Why do you need to play Qxg1, trading queens? White should keep queens on the board, as the resulting pawn race is lost.