Morph: Beast Boy: Give Beast Boy a move action that deals no pushing damage. After actions resolve, replace him with any character with this trait but a different collector's number on the same click number. Existing action tokens and action tokens for that action are applied to the replacement.

Tiny Little Arms: Beast Boy can't use the Carry ability.

CHOMP!: Give Beast Boy a close combat action. He can target up to three characters with a single close combat attack and divide the damage between each character hit. When he targets at least two characters in this way, he may reroll his attack roll once.

What exactly does "Give Beast Boy a move action that deals no pushing damage" mean? Can he use this trait if he already has an action token? Or does it mean that he can't use this trait if it will make him push?

Beast Boy can be given a move action whenever a character could normally be given a move action. If taking this move action would result in a second token being placed on Beast Boy, he would not take pushing damage for that action.

I've heard that there is some what that Beast Boy can move with Morph without actually changing characters... what's that all about?

The three common versions of Beast Boy (014-016) all possess a slightly different version of the Morph trait given above. The wording of that trait does not indicate that you must choose a different character, like the Beast Boy: T-Rex trait does (more on this in a moment). So a player was able to use the Morph trait for 014, 015, and 016 Beast Boys to move the character without fear of pushing damage and without needing to actually change the character.

However, this was not the intent of the trait and the next version of the Player's Guide will indicate that all of the DC75 Beast Boys possess the same trait as the 055 T-Rex version given above. So, no, you can not move Beast Boy: Bear with Morph to avoid pushing damage and continue playing Beast Boy: Bear. You must choose another Beast Boy witht he Morph trait.

What happens if Beast Boy uses Quake and Chomp! together? How is that damage divided?

Beast Boy can not use Quake and Chomp! together; both powers require that he be given a close combat action to activate and you can't give a character more than one non-free action per turn. You would need to choose whether to use Quake or Chomp!, not both.

However, since Blades/Claws/Fangs activates whenever you give a character close combat action, for any reason, you can use Blades/Claws/Fangs and Chomp! together to potentially boost Beast Boy's damage.