Welcome to the puppy world! First of all I would invest in a pen. I like the Iris pen that you can put a bed/crate inside. It gives the pup a little more freedom than a crate. I line the pen with a washable human grade pee pad (pharmacies have them. I confine the pup whenever I can't watch it.
As for the biting, she needs too learn that she can not put her teeth on you. Period. I used a hard rubber toy when the pup was in my lap. Only got it when she was on my lap. I would gently tap her head and say NO when she got too mouthy, and then I'd give her the rubber fish. Remember, this pup is teething, and NEEDS something to chew on. Occasionally when my dogs were teething, I would freeze a wet washcloth (rolled up) and give it to them. You have to be there and watch them, 'cause you don't want them to eat it!
As for potty issues, the pen will help. Patience! She is just beginning to have bowel and bladder control at 12 weeks. I don't take my chi's out in winter, so they 'go' on human grade pee pads. You might put one down near the door, IF you are going to have her go out to potty when the weather gets better. (I'm in the Chicago area)
As for the cat, I'd leave it up to the cat to take care of things. My cat has no claws, and jumped into the pens when my dogs were babies, and hit and hissed at them! They have a healthy respect for her now!