On interstate transport: Yup. Form 5320-20. Can be filed for any period up to 364 days (eg: 01-01-20 through 12-31-20). Required for all NFA items other than AOWs and suppressors.
On the Trust address: Different Trusts are worded different ways. Mine has never had a fixed address, only a declaration that it was governed by the laws of Vermont. That got changed such that I and the Trust share a common address, and it is governed by the laws of whatever state I find myself living in.
On purchasing toys in the state you don't live in: I don't know for certain, but I'm gonna guess no. The guess is predicated on the fact that you'll have to do a 4473 for some/most purchases. You can purchase a Title I long gun in another state under GCA '68. Handguns or frames/receivers, no. NFA,,, No bloody idea.
But... Any Trustee (responsible person) can purchase something on behalf of the Trust in their state. Including your out-of-state Trustees. IANAL, and all standard disclaimers apply.
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edit due to Windows crash.