I'm not sure I completely understand if this is a job you just started in September, or if the problems started or are being tracked from September to December. If you have been there for at least a year and are entitled to X number of days off, that's one thing. If you just started in September, and you have earned no time off yet (most companies don't give anything for 6 months), and yet you were out the 2nd day of school, that's a problem. If you have accumulated time off (sick, personal, vacation) and are giving advance notice when you can (vacation always, personal often, sick never), there shouldn't be a problem.
I have no idea why the supervisor (or you) would be counting the 4 days the office was closed. That's a paid holiday, right? So it's irrelevant to the discussion. So the only days to be discussed are the 5 remaining days when the office was open and you weren't there. Why she would say you were out 20 days, I have no idea. That should be a simple error to correct - you simply talk to her or to HR or the accounting office, whoever keeps the records. Ask for a list of the 20 days she has you recorded as being absent. If should be very simple to recreate your schedule and the days you were there. Don't take the approach that she's dishonest or aggressive, but that you all strive for accuracy in the record keeping. I think it would help us answer if we knew exactly what you are entitled to. For example, if you have been there at least a year, perhaps you get 5 sick days, 2 personal days, and 10 vacation days. So that's 17 days for a year, and you took 5 in the 4 month period from September 1 through December 31. So that's probably about right - 1/3 of your days in 1/3 of the year. But if you only started in September, it's absolutely way too much time because you haven't accrued it yet. Even if you take it without pay, that's not the point - you aren't there to work, your team has to work around you, someone has to bring you up to speed when you return, etc. It shows a lack of commitment.
I think it's not productive to worry about how much time off your supervisor takes, or how much she's entitled to. I'm guessing you don't have access to that information, nor do you know how many hours she works beyond 40 per week. You work part time and leave at 2:30, and you don't know if she leaves at 5, or 7:30, or works Saturdays and evenings. Moreover, what a person does on their time off is not relevant. So if you are home with a scared child or she is at a sporting event, they are equivalent in the eyes of the company. Neither one of you is there, and so no one is doing your respective jobs at that moment. I suggest that perhaps some of your attitude that her time off is not as valid because she is attending sporting events may be coming across in your dealings with her, and that may be taken as offensive or unprofessional. Likewise, whether your coworkers have children is irrelevant - you are all expected to be there when you are scheduled to be, and people with children don't get special consideration. If you or anyone else has a family tragedy, as your supervisor did, one would hope the company would be sympathetic. But my guess is that a child who's afraid to go to school and whose mother allows her to stay home all day is not going to be seen in the same category as a family death or critical injury.
As for the rescheduled meeting, there is no way for a large meeting with multiple departments involved to be scheduled at a time that's good for everyone. You had 4 days notice, which is plenty of time to find a sitter for a child. It was in the afternoon, so a high school student could have done the job I'm sure. Rescheduling an appointment with an optometrist shouldn't be a big deal. For a physician with a 3 month waiting list, yes, that's a huge hassle. But an optometrist shouldn't have that long a waiting time. In fact, if you had a sitter for your child so you could go to the optometrist, that should have covered you for the meeting as well. But these things happen. If your supervisor is in charge of this large group, then she gets to set the meeting times. Her athletic event is no less important than your optometrist in the eyes of the business world.
When you were hired, what were you told about staying past 2:30? Did they say they would give you notice? Did they say they wouldn't expect you to do so on short notice? What's in your employment contract or offer letter? Is there an employee handbook? Talk to HR and/or go back through your paperwork at the time of hire to be sure you fully understand all the terms and conditions of your employment. Then meet with HR and talk about ways to improve workplace communications so that you have a better relationship with your supervisor. Even if she is unreasonable or a poor communicator, you can learn to work within that scenario to have a more positive relationship.
If you can elaborate on some of these questions in your "so what happened" section, it will help us to understand the situation better and give you more advice.