Your hen is an American game hen. They are a breed, but a variable one. There are many strains of American games, but unless the breeder knows what they are, then it is impossible to know exactly what you have other than the general term, American game.
American games can be feisty, but the hens can get along fine with other hens so long as they are introduced to each other properly. Some fighting is to be expected. The chickens have to sort out their pecking order. This is true of all breeds.
American game hens will lay mostly in spring, then usually sit on a clutch of eggs to make chicks. They are very broody. If she does go broody, you can either break her broodiness by placing her in a cage that has bottom ventilation for w couple of days, or let her sit. You could probably buy a few fertile eggs if you wanted to let her hatch chicks. This would be a good way to introduce a friend or two without fighting, but then you run the risk of getting a rooster, which can be a problem.
You did a nice thing in taking her in, and she looks pretty happy as she is.
