I feel like I outplayed an Expert but still lost

I call h4 bad because I think in that structure, castling kingside, it's more appropriate to go for an f5 break... and if you play both h4 and f4 (as in the game) you have a lot of weak square around your king. Also h4 is just slow because standard KIA stuff doesn't work (I assume. I play KIA vs french sometimes where I'm not facing a kingside finachetto).
For example in a dragon structure where white castles kingside he goes for f4-f5 not an h pawn push.

I've played much higher rated players before and in that moment felt like I gave them a good go. Upon reviewing the games I find mistakes on both sides which I didn't consider during the game and more often than not I see areas where my opponent should've capitalized on my mistakes but missed it. I then realize maybe I didn't play as well as I thought. As I say though in the moment it feels pretty darn good. When you post games here a lot of experienced players will look at it and see mistakes, but it's still pretty cool that you gave your opponent a good go.

I still feel like discussing if White can draw the endgame.
8/6k1/3KPn1p/5p1P/5P2/8/8 w - - 0 1
It looks to me like Black wins this endgame. If I recall correctly from the book "Amateur to IM", the key squares for this h-pawn would be g1 and g2, so as soon as Black's king can get on one of those squares, he wins. Black must lose his f-pawn immediately after Ke5, so Black's h-pawn is his only hope to win. I discovered that the knight and h-pawn can set up a shield to keep the White king at bay, White cannot abandon Black's h-pawn or else it will run to promotion, and the knight can lose tempi and force White's king out of the h1-corner so that the h-pawn can queen. Therefore the only hope White has for a draw is to promote one of his two connected passed pawns if Black starts by ...Nxh5. It looks to me like through zugzwang White must lose both of those two pawns, but I'm not completely sure. Any other opinions?