SoCal,
On the first issue: employment.
Who paid you? If you were paid directly from the student then you are a sole proprietor. If you received payment from the school, and in turn received a 1099 at the end of the year, then you are an independent contractor. The key is - did (or will) you receive a 1099?
But seriously, we are splitting hairs here. Either way you will fill out a schedule C for your taxes and we are really talking about the same thing with regard to freelance flight instruction. Personally, I would list it as an independent contractor since you were representing the school, so to speak. Does that make sense?
The second issue: termination.
I agree with RickAir here... the arrest and the termination are two separate issues - according to you. So what's the story? What reason did they give you for termination? Non-renewal of contract is just that - not termination. So in order to know how to answer the termination question on an application it's important to know what the company said to you.
Yes, the arrest will show on your FBI report. And just like RickAir said, many airlines ask if you have ever been arrested. Be prepared to talk about it. It is great news that you were not convicted - that is hugely in your favor. If you have to disclose it on an application, please, please, please do NOT write "will discuss in interview." Nothing ****es off a recruiter/interviewer more than that. Have the decency to write a statement of what happened - they will respect you more for it. Once they read the statement they may decide it's worth diving into details a bit more and invite you to interview. If you don't write anything then they will automatically think you are hiding something and won't bother inviting you.
I would recommend obtaining a certified copy of the disposition of this case and keep it with your interview materials. This will be very helpful.
I wish you the very best,
Lori