Should I remove moulding before adding a plaster ceiling?
Yes, you should remove the trim before the ceiling is done.

Yes, some and possibly a lot of the material will get damaged if you have not done it before. Even a pro will crack some, but a pro will know how to repair it too. Sometimes it is better to get a new molding that closely matches the original.

Yes, the wall will get damaged too here and there but the same crew doing the plaster can mud in the damaged spots too.