Does anyone happen to know the name of the hotel that Qatar Airways uses for business class overnights?

Oryx Rotana.

Thanks Mike

Is this hotel located in the airport or does one have to travel back and forth from the airport?

There are two Oryx (at least two) hotels in Doha. The Oryx Airport which is within the airport and only available to transiting passengers and the Oryx Rotana which is offsite and requires clearing immigration.