Car not starting is considered a personal reason. You should have, according to general unemployment law and leaning, have had some form of back up way of getting to work, or alternative means of transportation to use at work.
The breaks issue had nothing to do with your firing, and thus nobody is going to consider it a valid issue when they are looking at the reason this previous employer terminated you.
This was a what sounds like unsympathetic employer who wanted rid of you and grabbed this as an excuse. If the schools were closed, and the situation was desperate and unusual, they had the choice of accepting your circumstances or using it as an excuse to terminate you. Nobody will care that it was your only means of transportation or that you had been late or tardy a lot of other times. This was not why you were fired. You were fired for not showing up on a very terrible weather day. You had not, in the past, failed to show up. You didn't have write ups for anything except perhaps one previous absence due to illness. You tried to give as much notice as possible, and explain the circumstances to your boss but were fired for this one offense. That is what you should say in your appeal of this particular situation.
Now, as to whether you are approved on the voluntary quit from your most recent employer, that's a whole nother kettle of fish. Good luck.

