Hey Rockstar,
in the time that I've taken to go through and annotate your game you've received a hell of a lot of good advice here!
As many have said, your problems in this game weren't pawn problems. You chose the Scotch opening, which is based on violent attack on the centre by White right from the start, and which can also be a minefield of tactical traps. Maybe better to go for quieter openings until your tactics have improved a bit; either that, or learn a few lines and principles of the Scotch.
Both you and your opponent made a lot of mistakes in the opening and middle game, but you came into the endgame winning only to misjudge the way to handle that advanced pawn (which you could easily have removed).
Your question about king safety was justified; your opponent could have made mincemeat out of your K-side several times, but chose not to do so. Castling within the first 10-15 moves is nearly always a must, and you left it much too late. By then you were winning anyway, only to blunder the finish.
Don't worry about him advancing central pawns once you've castled. You have just as many pawns as he does; they can be exchanged or used to block, and if your development is better than his they can also be used to attack. Just try to keep them solid and not isolated or doubled, and he'll run into a brick wall. And remember that they can't move backwards; be careful when advancing them.
Anyway, here's the game:
There is another saying , probably the best of all:
"Chess is not played with sayings"