My question is if she is successful in hatching them will the roosters go after the chicks?
My mature rooster doesn't. He assumes they are his babies. Sometimes he helps Mama take care of them, especially when she needs help, but usually he just ignores them. Immature cockerels are not mature roosters, but my broody hens keep them in their place.
Will the other hens or pullets go after them?
Mine hardly ever do. If they do, Mama whips butt. Sometimes she whips butt if she just thinks they are getting too close. You are dealing with living animals so anything can happen. You read stories on here where a rooster or especially a hen goes after the chicks and the broody hen does not protect her chicks, I'm sure that happens. But with three or four broody hens a year I've never seen that happen.
How successful is a broody hen at hatching a clutch of chicks?
You are dealing with living animals so bad things can happen. Bad things can happen if you incubate them in an incubator. Typically a broody hen can do as well or better than we can with an incubator, but with living animals you don't get guarantees on any one event.
Should i move her and the eggs?
What do your facilities look like? How big is your coop and how big is the run? What does your nest look like? If room is tight moving her is probably a good idea. If you have room in the coop and in the run it's probably not necessary but some people do anyway. My broody hens hatch with the flock in nests 2' and 4' above the coop floor. That height isn't a problem. But if the nest is so small the hen is sitting right next to the edge it might be. Photos of the coop, run, and nest could be helpful in accessing this.
I'd suggest you do not hatch those eggs. If you are going to let her hatch I'd gather all the eggs you want her to hatch and start them at the same time. That way you know when they should hatch and, more importantly, they should all hatch at the same time. You do not want a staggered hatch where hatch is strung out over days. Those are stressful to you and often not real successful.
If you are going to let her hatch with the flock mark those eggs, I use a black sharpie. Then every day after the others have laid check under her and remove any that don't belong. You can still use them if you remove them daily.
if you decide to move her and keep her locked up until she hatches you don't have to mark the eggs but still start fresh eggs, all at the same time.
This is my first broody hen and I'm wondering if I should break her or let nature take its course and let her be a mama.
Totally your choice. You will probably get boys if you hatch. Do you have a plan for those?
My husband said she did come out of the coop for short spurts so I guess she hasn't been held up in there being stubborn but I want to make sure she is OK to.
Before a hen or pullet even starts laying she builds up excess fat. That excess fat is what a broody mostly lives on so she can spend most of her time on the nest instead of out searching for food. She will lose weight while incubating but that is normal, it's fat put there for that purpose.
I've seen a broody hen come off of her nest twice a day for over an hour each time. I've seen a broody come off once a day for 15 minutes. I never see some broody hens off of the nest but I know they are coming off because they don't poop in the nest. A broody should come off on her schedule to eat, drink, poop, and maybe take a dust bath whether you see her or not. She's not being stubborn she is doing it the way nature intended her to do it.