I believe that mana spells cast from the physical plane affect things on the physical plane, and mana spells cast from the astral plane affect things on the astral plane
That's right. The core rulebook doesn't do a good job of covering topics related to the astral plane, and didn't clarify anything to my knowledge in Street Wyrd. I'm going to be using knowledge from previous editions, so there's a possibility I'm wrong but someone will correct me if I am.
While spells affect only the physical or only the astral plane, they have a form that exists on the astral plane. So, while an astrally projecting mage can't fling spells at opponents that are only on the physical plane, they can perform counterspelling.
Second question: Can area-of-effect mana spells affect runners that are only active on the physical plane? Like stun ball or mana ball?
Yes, if the spell was cast on the physical plane.
Third question: Can a projecting magician assign a target to a manifested spirit that is only present on the physical plane? Two explicit examples: can a Spirit of Men be instructed to heal someone who is mundane using healing spells? Can a Spirit of Fire be ordered to attack a mundane Runner?
It's not clear in 6th edition whether materialized spirits are dual natured or not. In older editions they are, and I play them that way, but Materialization doesn't state that it makes a spirit dual natured. Yes, a projecting magician can command a spirit to target something/someone that only exists on the physical plane. People and things are visible from the astral, just shadowy and insubstantial.
Fourth question: can a projecting mage use the clairvoyance spell? If so, does his view become normal or astral?
Yes, and he views the astral.
Here are some things projecting magicians can do:
Recon
Manifest, and communicate with their team about things on the astral
Counterspell
Fight opposing spirits and other astral entities
Summon spirits to:
- target things on the physical plane
- obey the commands of another team member
- sort of indirectly cast spells on the physical plane, by summoning a spirit of Kin and choosing the innate spell optional power and commanding its use