Morph: Identity Crisis: Give Prodigy a move action or ranged combat action. After actions resolve, you may replace him with any other character with this trait on the same click number. If an opposing character took damage during that action, you may instead deal Prodigy 1 unavoidable damage and then replace him with a character with the Morph: Changing Frequencies trait on the same click number.

Glimmering Guardian: Modify the defense value of adjacent friendly characters by +1 if they are equal to or less than Prodigy's point value and have not been modified by this effect.

Iron Idol: Prodigy can use Invulnerability and Super Senses.

Can I use Morph while also using powers that Prodigy might have? For example, he can use Charge on some clicks. Could I use one a powers and then Morph?
Depends. Activating Morph requires that you give Prodigy either a move action or a ranged combat action specifically to use this trait. You can't combo Morph with most other powers for the same reason you couldn't combo Flurry and Exploit Weakness: they are each activated by a separate action and you can't give a character more than one non-free action per turn.

Charge will not work, though. Charge grants a free close combat action; that is neither a move action nor a ranged combat action. There is nothing in Charge to activate this Morph.

Would I still get to move or make an attack if I use Morph?
Sure. Move actions always result in a character moving (one way or another); combat actions (both the close and ranged varieties) similarly always result in some kind of attack. So declare your action to activate Morph, move or attack, then switch your Prodigy after the action resolves.

When exactly does the Morph switch occur? Specifically, I am wondering about pushing damage and when I'd have to check for Willpower: does the character using Morph need to have Willpower or the character I change into?
Pushing damage is dealt as actions resolve. The swap doesn't occur until after actions resolve. Any push damage that might result from the action would occur prior to the switch; if you wish to ignore the push damage, the character that used Morph would have to have the Willpower.