1. You acquire a class III weapon in your class III friendly state.

2. You move to another class III friendly state.

3. What if anything must you do to keep it legal in the new state?

4. Also, if you move BACK to the original state, do you have to do anything to keep it legal?

Any Class 3 weapon (i.e., NFA firearm) requires notification and approval to the ATF (on Form 5) for transport across state lines.  The Form 5 will ask you to indicate whether it is temporary or permanent.

An exception allows a C&R holder to transport C&R Class 3 weapons across state lines without a Form 5.

Another exception allows an FFL to transport any Class 3 weapon across state lines.

Incorrect on two counts.  AOW and suppressors do not require any notification to transport across state lines.  The Form used is 5320.20, not Form 5 (aka 5320.5) (which is a tax free transfer of registration).