Uh, what?
The APU isn't an ETOPS requirement under the MMEL, but sane people include it as an ETOPS requirement under their own MEL.
Your question about the doors is weird. The APU or lack thereof has no impact on pressurization.
If you have a door showing open after engine start, the airplane won't pressurize period. That's why we have procedures to bypass a dodgy switch.
Hit the books Kiddo.
PB