Can White stop Black's pawns?

If the Black king is on h7 instead of g6, does anything change?

If it is Black's move, does anything change?

What if the Black king is on h7 and it is Black's move?

a. no, white cannot stop black's pawns.

b. yes, the start position for black's king changes, otherwise, no change in result.

c. no change in result.

d. no change in result.

edit: yes, draw for b.

All the positions mentioned are a win for black, except this one where white can draw:

This method wont work with the black king on g6. He can try to distract the king with h7 and then trying the same method, but the key difference is white's king will have to be on f7 to force the promotion. Black can then play Qa2+, swap the queens and run the e-pawn. 

Thanks Retguvvie98 (three correct answers) and to bonddiggity for correcting the answer to b.

Now, another question. Would this position qualify as a "key position" if we were limiting our number to the legendary 300?

what "key position"      "legendary 300" are you talking about?

"In Russian chess folklore it is said that there are 300 positions which comprise the most important knowledge which an aspiring player must acquire."
Rashid Ziyatdinov, GM-RAM: Essential Grandmaster Chess Knowledge (2000), p. 12.

Lev Alburt, Chess Training Pocket Book: 300 Most Important Positions and Ideas (1997) is also built on the basis of this idea.

To become a strong tournament player, you must indelibly carve into your chess memory a certain limited number of essential positions and concepts. (Alburt, p. 7)