Should an alternate armorer be signed for property book items?

If the alternate is in charge of a separate arms room they should DEFINTELY sign for, at least via sub hand receipt. It is NOT against regulations if the person signing for the arms room has unaccompanied access and all the prerequisites for maintaining an arms room (I have been an armorer for three different units, one of which was consolidated for the BN, and worked in supply). Your unit commander is signed for everything in the unit, but the accountable/responsible party (aka the primary armorer) should be on the hand receipt books for it...if that individual is not the one primarily responsible for certain items, they can (and should) be sub hand receipted down.

The only person on the property book is the commander. It can be sub-hand receipted to whomever is designated. Now if you're the only one with keys to that armsroom then you'd be the one. If the primary has keys to both he's the guy.

No. Never. The only person that is signed for anything that is not assigned out is the commander. The arms room is owned by the commander. Everything in there can be handreceipted or issued to elements in the TCB but the armorer is not signed for anything.