should I try to put her in the wired dog cage when she is sitting on the eggs or leave her in the nesting box until one hatches? Our nesting boxes are up higher, so I don't want any chicks to fall
You will find that we do this all sorts of different ways. I let my hens hatch with the flock and raise the chicks with the flock while others isolate them. It sound like you've decided to isolate her so I'll not go through my procedure with hatching with the flock.
We can remove the nesting box she is in and move it into the wired dog cage (they are big cat litter bins screwed in).
If you are going to move her I suggest you move her at night after it is dark and use as little light and commotion as possible. One risk in moving her is that she breaks from being broody and won't accept the move. That dog cage sounds like it should be OK. You need a nest, food, and water in there. You don't want it much bigger than that. A broody should know to not poop in her nest and mess the eggs up but she has no qualms about pooping in her water of feed. Arrange it so you can clean.
I think it helps them accept the move if you can make their nest fairly dark. You might consider that when deciding how to make your nest. Move her and give her a day to see if she accepts the move, then get the hatching eggs.
One issue I've had when trying to isolate a broody and her chicks from other chickens is that if the chicks can get away from the broody where she can't protect them the chicks are in danger from other flock members. When they hatch, the chicks can probably walk through the wiring on that dog cage but the broody hen can't. You might want to attach wire mesh around the bottom of that dog cage so the chicks can't get out.
Once hatched, is there something I need to do special for the chicks like different food available or something the hen needs?
Are you going to let the broody hen raise the chicks with the flock or are you going to isolate them from the flock? Again, people do this all kinds of ways.
You do not want the chicks to eat Layer feed. There are enough studies that show that if the eat too much calcium they can damage themselves. The broody hen is not laying eggs while she is broody so she doesn't need that calcium either. When my broody hens raise the chicks with the flock, I feed all the chickens either Starter or Grower with oyster shells on the side. The ones that need the calcium for their egg shells seem to know it and the others don't eat enough oyster shell to harm themselves. If you isolate the hen and chicks the hen can eat what the chicks do.
The only thing I do for the chicks is make sure there is food and water down low where they can get to it. I leave everything else up to the broody.
It depends some on how you are set up. She will take them where they can eat. She may show them what and how to eat. She may tear food up into bite sized pieces for them. When my feeders for the adults are up high my broodies have spooned feed out to the ground where the chicks can get to it. By two weeks the chicks are flying up to those feeders to feed themselves.
How long do I keep chicks in with the hen?
At some point the hen will wean them. She stops taking care of them. I've had broody hens stop cold turkey taking care of them in all aspects, most do it this way. I've had hens stop taking care of the chicks at night but continue to take care of them during the day. I've had hens stop during the day but take care of them at night. I've had a couple of hens stop at three weeks, some go for well over two months before they wean them. Each time with each hen is different. If you want you can separate the broody and chicks at any time and brood them yourself. Some people do that. Once again there is no set way or set time to do any of this.
Good luck and welcome to the adventure.
We have room for a wired dog pen in the coop; should I try to put her in the wired dog cage when she is sitting on the eggs or leave her in the nesting box until one hatches? Our nesting boxes are up higher, so I don't want any chicks to fall 
We can remove the nesting box she is in and move it into the wired dog cage (they are big cat litter bins screwed in).
