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

Shadowmeld: When Dusk occupies a square of hindering terrain, give him a free action and place him in another square of hindering terrain within 6 squares and line of fire. After actions resolve, a friendly character of equal or lesser points that was adjacent to Dusk when given the free action may be placed adjacent to Dusk but can't be given an action until your next turn.

Stealthy Stalker: Dusk can use Regeneration, Stealth and Super Senses.

Can I use Morph while also using powers that Dusk might have? For example, could I place him with Shadowmeld and then Morph?
Depends. Activating Morph requires that you give Dusk 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.

Shadowmeld won't work, though. Shadowmeld is activated by a free action and then you place Dusk. That's it. There is nothing there to activate Morph. Since it is a free action, you could always use Morph after you resolve Shadowmeld, but Shadowmeld itself won't help you with Moprh.

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 Dusk 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.

When I use Shadowmeld, does either character (Dusk or the adjacent friendly character) have to roll break away?
No. Shadowmeld places the characters; it doesn't move them. Break away is only rolled when a character tries to move.