If you apped a character with a dæmon, like from His Dark Materials or carried over from Distant Skies, would touching their own dæmon have any stabilizing effect, or would it be too much like, uh, touching themselves? What about someone else touching it?

Dæmons are the character's soul made manifest as a talking animal with a physical presence. Someone touching someone else's dæmon is always a huge deal, positive or negative, and has an effect on the dæmon's person.