First puzzle: Both players' only hope is on the kingside, so g5 after king moves is the only way to win. 

Second puzzle 1...p=Qc1 2.p=Qh8 Qa1+ 0-1

Did I get it right?

On the second puzzle, they are going the other way. I flipped the board to make it easier to play black. :0

First puzzle g5 is not the move. Black is playing to draw but white is playing to win, and g5 allows black to get the draw by the exchange.

These are pretty tricky but they use a very similar idea.