If a character had the spell Astral Armor quickened or tattooed to them how would this function? specifically, when the character astrally projects, would the quickened/tattooed spell protect his/her astral body or would it stay with his/her physical body?

It probably says somewhere in MiTS, but my feeling is that spells stay with your aura, so they'd follow you when you project. Not sure, though.

Should be. After all, you don't leave weapon focii behind.

Yes, spells and whatnot accompany the magician when he projects. The same is true for the "aural print" of fetishes and the like. It's detailed in the core rulebook, but I don't have access to it at the moment to give a page reference. I'm positive it's there, though.

I believe it specifies mana-type spells only.