Home is usually set where you power up / take off at, however there are plenty of users who can report that their misconfigured home position ended up with a drone taking off at full speed towards Shenzen or the 0,0 lat/long point off Africa.
Elevation can be determined by GPS, but a barometer works better.
There are pros and cons to both-- PCB mount baros don't always respond well in settings with lots of moving air and various other nearby components that get warm-to-hot during operation.
Keep in mind for either method you also have an accelerometer, so even if GPS (or a crummy baro) provides a messy signal for altitude you can average it over a given timespan and compare that against how quickly you are accelerating upwards or downwards.
Elevation can be determined by GPS, but a barometer works better.