If I hit APP with the switch set to GPS what approach will the plane fly, if any?
How do I fly a GPS approach when available in the ATC menu?
The plane will not fly anything if you press APPR with the GPS/NAV toggle switch set to GPS.... The APPR mode is not like the NAV mode - it does not slave to GPS at all... APPR only reads Localizer and Glideslope signals from the NAV1 radio (and NAV2 radio in a Cat 3 armed plane).
As you've rightly said, GPS approaches do exist (you can access them in the PROC page of the GPS), to follow these make sure GPS/NAV is set to GPS, and press the NAV button.
The plane will slave to GPS and fly the approach - remember, you have no Glideslope assistance so you'll need to look after your descent, and also the GPS approach is "non-precision" so it is not guaranteed to line you up accurately with the Runway - so you should disengage AutoPilot at 3 miles from the Runway at the Latest and visually remain 'lined up'.
GPS Approaches are good fun though - mainly used in GA aircraft that are equipped with good modern GPS systems.... Airliners would be using ILS in most cases and using full precision NAV approaches with Localizer (line up) guidance and Glideslope (descent) guidance - although not always using the AutoPilot - a lot of pilots will fly in manually but ill use the ILS signals to make sure they are approaching accurately.