Suppose I play Draw 2 and Use ‘Em and I draw two Goals, and one of them will cause my opponent to win. Can I play the two Goals in instant succession, with the one that doesn’t make them win on top?

No. Any time a Goal is played, it provides a chance for someone to win with those conditions. No matter how briefly that Goal is in play, it counts, and if someone meets the conditions in that instant, they win.