a) Don't know what you mean... Sorry. You'd probably have to rip the phone all apart to attempt to fix this and from what I've heard here at CB that is one of the most difficult parts of the tour to access/take apart because of the flash as well as the top mute and lock buttons. Its all held together up there with some sort of tape. Wouldn't recommend breaking in there unless you really know what you are doing and are quite comfortable doing it for such a minor annoyance.
b) Some have placed a folded piece of paper or a piece of tape up under the back battery door and have had success on this one. It won't be so "floppy" for lack of a better word after this either.
yes a case will help with the issue of the door- if it becomes a large enough issue to where it is falling off I say ge a new door or ask for replacement.