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

Doing the Things a Spider Can: Spider-Man can use Leap/Climb and Super Senses.

Here's Web in Yer Eye!: When Spider-Man hits an opposing character, its range value becomes 0 and is locked until your next turn.

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.