Can I deduct mileage on a rental car used for my business, if the rental costs were covered by insurance?

No, you cannot deduct the standard mileage.  This accounts for general wear and tear to your asset and since it is not your asset you cannot deduct it.

You can, however, deduct the gas you put it in.  You can also deduct the auto insurance that you purchased with the rental. 

I understand I can write off gas expense for a rental car used for business. What category does this expense fall under?

Business expenses? 

You can either enter it under rental costs or make up your own category at the end of the expenses.