Are all apps compatible with the Apple TV? I can't find a list of Apple TV products anywhere. I want to use Dish Anywhere specifically on the Apple TV. I'd also like to know if I can just mirror my screen from my iPhone 6 Plus to the Apple TV.
No, not every iPhone app will be on the new Apple TV. Developers have to release apps specifically for the Apple TV, not just porting the iPhone app.

Yes, you can mirror the screen of any iPhone 4s or later to one of the newer Apple TVs.