Ahhh door templates...how about the Head Height parameter that shows you the height of the door and not the frame? That's so residential.
Be careful with starting from Generic Model and changing the family to a door. Some parameters that are available in a door family template don't show up when you use this method. As an example, you won't get a level parameter. I got burned with this (thank goodness I discovered this before the drawings went out). I had overhead coiling grilles built out of a GM family and turned into door families. My schedules were filtered by Level. Since these did not have a Level parameter, Revit kicked them out. My solution was to filter by name and not floor (Ex: A2 for area A, second floor, etc). Support told me to build doors using the door template
But that's really not the solution....they need to fix this quirk.