Morph: Identity Crisis: Give Hornet 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 Hornet 1 unavoidable damage and then replace him with a character with the Morph: Changing Frequencies trait on the same click number.

Sedative Stingers: If an opposing character's printed defense value is lower than Hornet's printed defense value after that character takes damage from Hornet's ranged combat attack, give that character an action token.

Cybernetic Carapace: Hornet can use Toughness and Super Senses.

Gadgety Defender: Hornet can use Perplex. Once per turn when he uses Perplex, roll a d6. On a roll of 5-6 he may use Perplex again.

Can I use Morph while also using powers that Hornet might have? For example, he can use Running Shot on some clicks. Could I use one a powers and then Morph?
Depends. Activating Morph requires that you give Hornet 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.

Running shot could work, though. Running Shot grants a free ranged combat action. If you give Hornet a power action to use Running shot, you could move Hornet and then, with the free ranged combat action, activate 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 Hornet 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.