Morph: Spider Armor: Give Iron Spider a move action or close combat action. After actions resolve, you may replace him with any other character with this trait, the Morph: Identity Crisis trait, or the Morph: Changing Frequencies trait on the same click number.

Gliding to the Rescue!: Iron Spider can use the Flight ability and Running Shot.

Three "Waldos": Once per turn, Iron Spider can use Flurry or Perplex, and can only use Perplex to target himself.

Can I use Morph while also using powers that Iron Spider might have? For example, he can use Running Shot on certain parts of his dial. Could I use that and then Morph?
Depends. Activating Morph requires that you give Iron Spider either a move action or a close 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 specifically will not work, though. Running Shot grants a free ranged combat action; there is no move or close combat action given here. There is nothing to activate Morph in this case.

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 Iron Spider 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.