The first puzzle was easy.
The Best and Most Difficult Chess Puzzles Ever

Re: Puzzle #2 - promotion to N is not required . . .
but then they do not need to take the knight with the king

ok, going to give it another shot. This is a puzzle by Richard Reti, I took from a book. I still don't get it. Is the puzzle correct or does it have faulty moves by black?
You made a mistake
3rd move should be Ke6 or Ke7
Ke5 loses because after h2 and Kd6 and h1=Q, notice that Black has a closer King to the pawn and it is not on the 7th rank, so there might be a mistake with that.
Correct it and resend it.

Re: Puzzle #2 - promotion to N is not required . . .
If you promote to a queen then Bxg2 and white is winning but missed his chance to win quicker, so the puzzle would not be solved

Puzzle 1: Prepare to kick yourself if you use the 'Help' button:
Puzzle 2: I'll give you a hint on this next one, the first promotion is a knight, you'll see why!
Puzzle 3: Another strange occasion...
Puzzle 4: Black had agreed in this game to mirror whites moves. Bad choice! Mate in 4!
Feel free to comment and post your own 'most difficult' chess puzzles.
Puzzle 4 has another solution
...
White to play and FORCE Black to mate White.