You'll need to file the 5320.20 at least 30 days before you move, I'd do it ASAP prior to the move date. Once the form is approved, you are good to go. The weapons can be stored in secure location, such as a safety deposit box, if need be prior to getting the approved form.
If you wish, send a letter notifying them of the change of address.
There is no official form for this purpose because there is no official requirement to do so.
Not sure this is a good idea. The trust is set up under the state of jurisdiction at the time it's set up, and it's it's own entity and can stay right where it is. I think you can change the mailing address for the trust to anyone you like, any time you like, but I don't know that there's a need to amend the trust, and there's a chance that doing so might render it invalid. Check with a lawyer before amending an NFA trust at all.