Egg cartons are safe to use during hatch. Cut out the bottom of each little compartment in the carton for air flow. I'm assuming you use a turner for the first 18 days.. if you only have one incubator, and have a still air, you need to keep the eggs at the same height that they were at in the turner, anyways. Using egg cartons is a good way to do this.
It's okay for the eggs to be rolled around during hatch. My preference is to lay them on their sides, it's never affected my hatch rates.
Leaving them in the egg turners is a bit risky. Chicks have successfully hatched in them before, but it's not ideal. There's too many places for toes and legs to get stuck and injured.