Any ideas? This is also my first time being the president of something, and I have no leadership skills whatsoever. I'm not sure how to act/talk?
Just be yourself, acting like you're playing a role would feel wrong in the long run and it's better to express yourself naturally. About leadership, while charisma is a plus it's not necessary. In fact, you could learn leadership abilities by having that kind of responsibility.
How do I make eye contact with the audience?
If you're speaking to an audience, you can stare roughly around where most people are sitting/standing. Make your gaze travel across the room as you speak, looking at different persons briefly as some become embarrassed if stared at for too long. It will come naturally so don't worry too much about it, it also comes with practice anyway. Just do whatever feels more comfortable to you, it's no use if you're too nervous yourself right ? ^^
How can I quiet members down if they're noisy?
If they're hardcore Manga/Anime fans, act like a tsundere ! ......... ^^'''
Um... seriously though, it's hard to find a balance. Your tone needs to attract attention but it doesn't have to be too harsh either. I'd say it depends on the situation at hand.
Should I create a set of rules for people to follow?
For sure ! It's always good to have a guideline on how everyone should/shouldn't behave.
What should be the extent of someone's rude behavior before kicking them out of the club?
Lack of respect for either you or other members on a regular basis, the usual stuff really.
As for ideas...
Hmm... None come to mind yet..