Nevertheless as far as point 2 is concerned, even though such a GET is technically valid in the eyes of halacha, it may have been halachicly wrong to have been done, and there even may be penalties involved for having done so. (i.e. if a husband violates Chrem Rabbeinu Gershom prohibiting him from divorcing his wife against her will. Even though the divorce he gave her was halachicly effective.)
A beis din is the one that completes the process of giving a get does not only make sure that the get itself is kosher but also that it is halachicaly acceptable to be done.
Btw, Cherem drabeinu Gershom was only valid for 1,000 and I think the 1,000 years are over already.