Spirit Healing
They can't. Well, a Spirit of Man with the Heal Spell or a Task Spirit with First Aid/Medicine could, but not generally. Really, the only thing those categories mechanically tie to is a bound spirit's ability to Aid Spellcasting/Alchemy/Rituals.
They fucked it up with a line in Street Grimoire that says that spirits of each tradition will only perform tasks related to their category, which left everyone wondering how (for instance) an Earth Spirit is supposed to heal. Some people came up with some really round about way of approaching it (i.e. a spirit ordered to defend is sort of healing, etc). I'd just ignore that line and go with the current mechanical effect. If its not enough, then make the spell sustaining/spell binding tied to the category as well.
But yeah, one of the few things you can't do as an aspected summoner is magically heal. Breeze actually has an alchemical healing spell attached to him, but it won't do anything unless he stabilizes (assuming he isn't dead-dead, although I imagine the extra bullets fired his way may insure he's dead-dead-dead).
RE Astral Perception: Oh yeah, that's right, I guess you did perceive at the beginning of the fight. Astral Perception is a little wonky. Can you still perceive in AR if you're astrally perceiving? I'm not sure, it's not really the physical world... Or whether you would have taken the penalty to spotting Dantes. Did you ever assense his aura?