Is she too young?
Physically she is old enough. Some first time broodies have a little trouble figuring it out since it is all on instinct the first time regardlessof age, but the instincts are probably pretty good. If you want chicks, I'd say go for it.
How many chicks am I likely to get from just 6 eggs? (I've found a local place where I could get some fertile eggs and they say they'll take any resulting cockerels off my hands.)
Are you asking how many eggs will hatch (I think so but not really sure) or how many female could you expect if 6 hatch? Or are you asking how many eggs to set to get 6 females? If the question is the odds of male versus female, here you go.
6 male 0 female - 1 in 64 or 1.6%
5 male 1 female - 6 in 64 or 9.4%
4 male 2 female - 15 in 64 or 23.4%
3 male 3 female - 20 in 64 or 31.3%
2 male 4 female - 15 in 64 or 23.4%
1 male 5 female - 6 in 64 or 9.4%
0 male 6 female - 1 in 64 or 1.6%
If you are asking how many will hatch, it depends on so many different things, it is very hard to tell. Some of the things it depends on are the nutrition, health and age of the parents, the quality of the eggs, how they are handled, and how well your broody does. If they are shipped in the mail, I think a 50% hatch rate is pretty good. If you pick them up yourself and handle them carefully you can improve your odds but you need to be ready for any result between all or none. If you want 6 female, I'd set 12 eggs.
Would an old rabbit cage be suitable for a brooding cage?
Rabbit cages are normally all wire construction and probably elevated. It mainly depends on the size, but it should work well if it is big enough. You'll need room for the nest and enough room for food, water, and room for her to leave the nest and go poo. How much room you actually need depends on the size of your feeder and waterer. Broodies tend to like a darker nest, so I'd be tempted to close off at least two sides (maybe 3 if you can easily) and the top of the nest to make her feel more secure. The nesting material will close off the floor. Since she will spend very little time actually on the wire, I would try to keep the nesting material in the nest and leave the other wire open. The poop will not fall through it but it will be easier to clean and provide good air circulation.
How long until the new chicks can be introduced back to the other girlies?
Opinions vary all over the board on this one. I'll include a link to a thread that discusses this, although I'll admit this particular thread mainly supports my point of view. What I would do is set up the broody pen in the coop so the hens are never really isolated so there is no real reintegration later. Then, when the chicks are a day or two old, open the door and let mama raise them with the flock. There are risks involved with this but there are risks involved with any approach you take. It is my personal opinion (and I respect the opinion of others that do it differently), but with my set-up, philosophy, and goals, this is the approach I choose to take. If I had less room, I'd possibly think differently.
https://www.backyardchickens.com/forum/viewtopic.php?id=215937&p=1
Good luck!