1) No you should not remove the battery. No reason to. If the system get so hot that it can effect the battery then the rest of the system will die before the battery. Heat is the killer of all things Electronic.
2) Yes. Read # 3
3) Yes by running the system on AC only without a battery inserted and if the power goes out the whole system could become corrupted. So don't remove the battery. No good reason to.
4) I have no suggestion.