For us it's the MEL until we block out prior to take-off and after that it is QRH. So for your 'doors closed' scenario it would be MEL until block out and that would involve 'doors open' again and having the MEL signed off by the engineers; we cannot apply our own MEL and do our own paperwork.
We do have a useful EICAS cross reference list that shows the related MEL that applies to an EICAS message which can help in making the decision whether to go or return to the gate once we have blocked out. For instance, AIR/GND SYS in the QRH deals with the airborne case but it does not say 'do not take off'. The MEL says that but, strictly speaking, it is not a requirement to check the MEL after block out (I'm not saying that I wouldn't, in fact I almost certainly would). The cross reference table tells me that it is a no dispatch item so that's a pretty simple decision in my mind. The table is located at the back of the QRH, so it's easy to check having completed any QRH actions and realising any implications of the EICAS message you've just dealt with.