Oh look a few questions I can answer! I don't know what age a hen must be in order to brood, but I would guess past when they start laying eggs. There doesn't need to be a rooster. I've got a broody Australorp right now who hasn't ever seen a rooster in her life.
She needs to sit on fertilized eggs if you want them to hatch. Unfertilized eggs won't hatch. But whether the eggs are fertilized or not has nothing to do with whether or not the hen goes broody. She doesn't have any idea, she just wants to sit on some eggs. That said, I wouldn't let her sit on unfertilized eggs. Those are eggs that you're not eating, you run the risk of a rotten egg exploding under your hen, and brooding is stressful on the hen, so there's not much point in letting her go through it without getting some babies at the end. There are many ways to "break" a broody hen, some nicer than others, but are generally better than letting your hen starve herself in a futile attempt at hatching babies if you're not going to give her fertile eggs.
I've heard of people having success letting a hen sit on eggs for a couple weeks, then slipping some chicks under her at night. I've also heard quite a few horror stories of adoptions gone wrong. So yes, it can be done, but success is not guaranteed. Of course, not all broody hens make good mothers, so success is not guaranteed even if she hatches them herself.
