Can I move second-floor duct-work from the floor to the attic
Drywall is the enemy.

That dryer vent is disconcerting; lets find out what other shenanigans are going on here. You may not have ever done ductwork but at least you seem to know a proper installation will require opening the walls to install rigid (use flexible only if you must).

Can you run it in the attic, sure: at a loss of efficiency; longer run, more leaks\back-pressure. Also, efficiency will drop like a rock if you run any ducts through unconditioned space. Ideally everything is within the building envelope.

Running ducts isn't that hard; fixing what you had to destroy to get them in, is. 

You will cut yourself while working with sheet metal. Proceed slowly and methodically; everything is sharp...