Leonardo
(1) I believe that engineers in transports have to dismount to perform an engineering task
(2) Engineers can certainly demolish buildings (p34 of rules) ...so we play that this means they can demolish BUAs. We play this means a 5cm x 5cm area of BUA is demolished and all units in it are destroyed.
(3) That's what the rules say. You could add a house rule though.
(4) I'm sure Pete has confirmed that when clearing mines..engineers can 'move up' during the clearing initiative phase. This allows them to continuously clear in each consecutive initiative phase to get though a minefield.
Hope this helps