No matter how much they mate you will not get chicks unless you first get eggs. If she has stopped laying due to the molt, she is not likely to lay any eggs until after the molt is over. She may start laying soon after the molt is finished or if you are north of the equator she may wait until spring and the days get longer.
If the EE rooster is an adult, he will probably mate some with the hen to establish his dominance, whether she is molting or not. What happens could also depend on whether you have any other hens in there, or even another rooster. What other chickens you might have can have an effect on what happens. If he is an adult rooster they will probably get along great. Hens send out signals that they are ready or not ready for mating and adult roosters usually respect that. An immature cockerel is another creature entirely. Things could get pretty wild if you put an immature cockerel with her or other hens.
Once she is laying, it is not a case of if “they bother having chicks”. To hatch the eggs you either have to have a broody hen or an incubator. Not all hens go broody and they certainly don’t go broody at a time convenient to you. If you have other hens and one goes broody you can give her the eggs to hatch. Hens will attempt to hatch other hen’s, turkey, duck, or pheasant eggs or door knobs. All some need is an imagination. It does not need to be her own eggs.
Having a rooster around has no effect on whether a hen goes broody or not. The only way you can control when eggs hatch is to use an incubator. Other than that, you are just depending on blind luck and that usually doesn’t work with chickens going broody.
