I am planning to submit an abstract for a potential talk at a school. I was wondering if I should use my university's letterhead for the abstract? If not, is there a specific format that I should use?
I wouldn't bother: if the school at which you're talking wants to advertise it, they will want to put it on their own letterhead to avoid confusion. They will probably reformat it to fit whatever their in-house style is, so it's best to just write it in a standard plain-text format (judging by your usename, LaTeX).
It depends, if you are doing this through or as part of your university association / studies then yes and that is fine.
If this is in your own time and and based on a hobby external to the university then maybe no, unless you want to mention “in passing” that is where you currently work.