I laid mine directly on the screen, not knowing any better. Some people say that the egg carton makes it hard to keep the humidity steady, cuz the box absorbs liquid. Rancher does staggered hatches, I think. But you are right, you will be opening it twice a day to turn them quickly, so you can add eggs to the 'bator. If you have a separate brooder to put the first hatched into, then you shouldn't have any major problem with different hatching times.
I want to say "sucker", but I have chickens because my Mallards from the previous year didn't come back....so your "excuse" is far better than mine. Enjoy your flock. I am sure your daughter will love having the chickens around.
I've used paper egg cartons after rinsing with hot water and letting them dry. I never wet the cartons. I've use the styrofoam cartons too, but I don't put them in cartons until lockdown when there is not turning to do. My hatches always take three days to complete at least. The start on day 20. I do keep the temp above the 99.5 also. 100.0 - 101.0
You really have to practice and get your own system and routine down.