Morph: Spider Armor: Give Spider-Man 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.

Anti-Sinister Six: Spider-Man ignores Plasticity and Shape Change and can't be targeted with Outwit, Penetrating/Psychic Blast, and Perplex. Spider-Man ignores damage dealt by characters who were given an action to use Charge this turn. This trait only applies to characters less than Spider-Man's point value or possessing the Sinister Syndicate keyword or team ability.

Can I use Morph while also using powers that Spider-Man might have? For example, depending on the version, some have Flurry, Charge, or Leap/Climb. Could I use one of these powers and then Morph?
Depends. Activating Morph requires that you give Spider-Man 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.

Charge could work, though. Charge grants a free close combat action. If you give Spider-Man a power action to use Charge, you could move Spider-Man and then, with the free close 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 Spider-Man 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 Spider-Man using Morph need to have Willpower or the Spider-Man 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 Spider-Man that used Morph would have to have the Willpower.