What happens if your character tries to possess a possessed being? Furthermore,
what happens when your character is using channeling, and someone tries to possess him?

I have no idea how to handle this by the book. But I'd have the two possesing 'people' compare successes. Who ever possesses the poor bugger better is the one who actually gets him. If he's already possessed and you want to possess him then I'd run it as if you were possessing the one who possessed the target first.
I really need to just make names for people when I do things like this. I'm not certain I follow what I just wrote.
As for Channeling, if this is where character A is contacting the dead, or some such then if the possesion is successful I would have the channeling cut off.

Channelling is a metamagic from Target: Awakened Lands that allows an initiate to draw a spirit into himself and remain in control, gaining physical bonuses and the use of the spirit's powers.

Personally I'd say that a posession would not oust the spirit from the body, but that the bonuses&powers from the channelled spirit would not be available to the posessing being. If the posessing being was somehow ousted then the original character would take control again and the bonuses and powers would be available again.

I don't have my books on me right now, once I check them the above may be revised.

I'd say that since in order to possess someone you have to defeat them in astral combat, you'd have to defeat the initial possessor that way, and if you win you snag the meat body.

As for the possession while channelling, I'd say since they're actually trying to oust the mage AND the spirit, but since the spirit uses all its services in order to be channelled in the first place, you should stack the force ratings of both for the new arrival's target number for astral combat, but the channelling mage only gets to use his basic magic rating (sort of the spirit makes it harder to oust the mage from the body, but doesn't actively help the mage in question other than that cos the spirit doesn't owe the mage anything any more, its services having been used when it was channelled).

my take, anyway.