When doing circuit work I've always used QFE but as it only relates to the ground height of a specific point (the airfield) then that is the only time it is useful. As soon as I'm out of the circuit, it is QNH.
GPS height readings can't be trusted unless you are using a device which has SBAS (like EGNOS or WAAS). The accuracy can be all over the shop and it also depends on which Geoid is in use as well as how accurate the terrain database is. At best it might be OK as a gross error check.