The All American Game

I've caught the last 3 games in each series leading up to the World Series, and am by chance watching the 1st of this series now. I'm not much of a baseball fan, and don't understand it quite as much as other sports. It's the one sport my dad doesn't care for so I never grew up with a game on in the background. I could hold my own in a game of hockey, football, or basketball, but I'm still learning in baseball.

3 things I learned about the game this week:
1. The American League pitchers don't bat. Still haven't figured out if this is just a rule when they play other AL teams, or if it's the case all the time. If so, when AL plays an NL team, do the NL pitchers not bat, or do both bat?
2. A batter can run for a base if the catcher drops the third strike. Now, not certain if this is always the case, or only when 1st is free. It would seem logical only if the base was free, otherwise the fielding team would have a force out throw at second. But I'm not sure.
3. The GM for the Cardinals bears an uncanny resemblence to Pete Rose.

I've heard quite a bit of flak about the last one. It's usually divided along gender lines with women agreeing with me, and men telling me that I'm nuts. That could be because the only men I've asked were my dad (who lived in Cinci and was a Pete Rose fan even though he didn't watch the game much) and my brother-in-law (who is a diehard Cardinals fan). Maybe I should ask someone more objective.