that tire is done in my book. i'd get a new tire. :)

make sure the tire shop you take it to knows how to mount on 19" wheels with TPMS without damaging the finish. try your local speed shop that sells and mounts tires on volks/weds/bbs/advans, etc first. DO NOT trust just any tire store. they will damage your wheels.

That tire is a write off. The screw is in or near the side wall. You need a new tire, I'm sure most shops would not patch that. Sucks cause these skins aren't cheap!

I'd get a new one, dont risk on the sidewalls

What is the trick to mounting them? What is difficult? How much should I expect to pay?
I already ordered a new one from tire rack.

I am thinking I will just take it to Nissan and have them change the tire. Does that seem like a good idea? I figure that if they screw up my rim I stand a better chance of recouping my loss than I would from a tire shop.

Yah, check out the other thread wat can happen to the wheels when incompetent tire shops mount using prybars and traditional mounters. I think your dealer should be able to mount without scuffing them. Typically shops charge 20-35 each which includes mount, balance, disposal of old tires. :)