Yes.
Yes, you will do this in Terminal A. In theory you can do it in F, but to do that you need to exit the airport after your re-check your bags and walk outside all the way to F, which is a long way.
The largest AA lounge is in Terminal B/C (entrance is located in the connector corridor between those two terminals). You can walk there airside from Terminal A, 5-10 minutes. I am not positive that there are showers, but if there were to be showers in any AA lounge at PHL, it would be that one.
Yes.