Form 1 is an application to make and register a firearm. The ATF considers any assembly or alteration of a weapon to be "making" a weapon, so converting a shotgun to a short-barreled shotgun would fall under this category and will require a form 1.
Form 4 is an application to transfer firearms from one person to another. Either of the "people" involved in the transfer may be individuals, corporations, trusts or any other legal entity able to own weapons. Most transfers require a form 4 to be filled out even when transferring a weapon owned by you to a trust created by you.
Form 5 is an application to transfer firearms from one person to another without acquiring a tax stamp. The typical case for submitting a form 5 would be inheriting a weapon from a deceased relative of whom you are the lawful heir. Other use cases include unserviceable firearms being transferred as a relic or ornament.
Form 20 is an application to transport firearms from one state to another. This form is used for a permanent change of address or for weapons which will be temporarily relocated to another state. Generally, moving from one state to another will not require any changes to the trust document. However, the weapons must be legal in their destination. Note that Maine does not allow trusts created in another state to purchase firearms.
Every responsible person in your trust must fill out a Form 23 when submitting a Form 1, Form 4, or Form 5 to the ATF. Form 23 provides identifying information to the ATF and each responsible person's chief law enforcement officer.
Although it is still illegal for non-immigrant aliens to ship, transport, receive, or possess firearms except for in certain special circumstances, this document has been rolled into all applicable ATF forms. Thus this form is no longer required and should be considered deprecated.