I was fired for driving a company vehicle without a license, I filed for unemployment and it was denied. Before being fired I was put on suspension and I wrote a letter asking to be transferred to a position that doesn't require driving and was still fired. Please advice on how to present my case on a hearing?

Company was not require to give you a transfer,
If you did not have a license then you needed to tell them before you you drive and don't drive, before you were caught driving without a license is when you tell them and ask for a transfer.

You file an appeal by telling the unemployment office that you want to appeal

It is your right to file. I doubt you will have much of a chance

I was awarded unemployment compensation. If my company appeals and wins, do I have to pay the benefits back?