OK, since you were nice enough to number, I will too.
1. She ran at the last minute - so what? Apparently more people wanted her to be president than the other girl - who I'm kind of guessing thought she had it in the bag. IMO, good going. Nothing worse than someone who thinks she has a mandate just because no one else wanted to step up.
2. I've heard of people being elected to an office literally a week after they were initiated. If it's true she doesn't know the rules, is anyone trying to help her with that? Or are they just criticizing?
3. I'm not sure if the "she" who made mistakes is the prez or the rush director so I won't comment on how the subsequent fines should have been handled. Either way, no, the prez should have not let someone who didn't make grades have an office, but here was her choice: kick someone out at the last minute (I'm not sure of the timing here) and thrust someone into arguably the most important office in the sorority who had no clue what the eff they were doing, or bend the rules and get through rush. How did you do in rush - did you make quota? Are you at total? Are you happy with the new members from that rush? If so, then quit complaining and move forward.
4. I'm going to be straight up here. You sound like a jealous beeyotch. Has anyone talked to her about her alleged drinking problem? Or is it just an issue because she's drinking her beer out of a can instead of a glass? Do you have anything in your chapter bylaws regarding punishment for attending unregistered events?
I also have to ask...how big is your house? If you have 40, and the house holds 20, I hate to break it to you, but her "housemates" are half the sorority.
The complexion of your chapter sounds as though it's changing. I'm sorry if you have a problem with it, but you can either go with the flow or make yourself miserable and get a reputation as a jerk.
ETA:
Be careful what you wish for, or you will surely get it.