1) If I attack rooted units, is there a fight

2) If there is, why is it

3) If I attack rooted units, with possible support, is there a fight (and what happens if i Win/Lose)

1. Yes. (See the AGoT 2nd-Ed. FAQ v1.4, where this is stated explicitly.)

2. Because FFG says so.

3. See the answer to #1. Combat takes place normally, so support is calculated normally for both of you. If the attacker wins, the defending routed units always die, regardless of the presence or absence of sword, tower or skull icons. If the attacker loses, they may have to retreat or their attacking units may die, depending on the sword, tower and skull icons on the cards that were played by both players.

What happens if I attack rooted units, on a field where they have a power token? They have support, and I have lost the fight.

Is power token still there, confirming loyalty for losing side, or not?

I have looked for this in Rules Book, and FAQ. Nothing on this problem.

If there are routed units on a spot with a power token, and you attack but are defeated, the power token will stay in place. Power tokens are only removed if land changes from ownership of one player to another. If the attacker has lost the battle then there the land never changed owners.

This is the same if you are attacking non-routed units on a power token