If a hero or monster shuts a door, does it require a skill test to open again? Or is it merely treated as a normal door from then on?

Also, while it seems unlikely to happen, the alternate victory condition for the heroes is to 'move off the map' - do all the heroes have to do this to secure a win without killing Eliza?

If a hero or monster shuts a door, does it require a skill test to open again? Or is it merely treated as a normal door from then on?

This comes from the FFG Sez rulings on BGG:

Q: In Encounter 2 of "The Masquerade Ball," would a door which was closed after it was opened require a test to smash/answer riddle/pick lock again? Or can the door be treated like normal? Similarly, does the stuck door in "The Shadow Vault" require Might testing every time to open it?

A: It would appear that there are a few different answers to your question here. For the doors in "Masquerade," a successful Might test removes the door from the map entirely, meaning it is set aside and cannot be closed again. Obviously, the Knowledge test does nothing with the door itself, but the Awareness test only opens it, meaning it's free to be closed again and would require one of the tests to get through. The same applies to the door in "The Shadow Vault," but in this case the Might test only opens the door, leaving it able to be closed again and thus requiring another successful Might test to open.

Indeed. There are some doors ibn other quests that are locked and then become normal after they are opened or a trigger is fulfilled. These will say "...is now treated as a normal door" or something like that.