Every now and then you see a goalie scoring a last minute goal on a corner kick. I remember being at a game in Mexico and having a goalie famous for his relative offensive skills heading one in to tie it up, but then he got beat with a shot about 90 seconds later and lost the game.
As for the "man advantage" it isn't that rare, with a red card team goes down a man. Its definitely a plus, but teams don't necessarily dominate with it. When it gets to 2 or 3 men, then you start seeing teams able to really dictate the game. That doesn't happen too often except very near the end of games.