These are kind of nebulous issues in several respects.

Employment: The official definition of "employed" would probably require a W-2. If you did not do the W-2/W-4 drill with the school, then you were a subcontractor. The reality from an interview/background check perspective is that they are going to want to call someone and verify that you worked there. You want to make sure that whoever they talk to tells them the same thing that YOU told them. If in doubt, work that out in advance with your reference. If you had a W-2 and did freelance, then you were both an employee and a freelancer. You will need a reference for the freelance work..another CFI or even a student just to verify that you weren't running dope from mexico during that time frame (I'm serious).

2. Not sure why you got fired if it had nothing to do with work? If work knew about the arrest, then a potential employer might hear about it when they call references. The good news is no conviction. The bad news is that some (or most?) airlines can and will ask if you have been arrested. If you lie and they find out, expect to be terminated and pretty much blacklisted from 121 aviation.

The last one is a tough call...in the current climate anything questionable could easily cause an employer to move on to other applicants with clean records (less hassle for the HR folks). I would probably try to research what specific employers ask on their apps and interviews...hopefully you can find someone who does not even ask about it.