1) You can get the cat fixed, for an inexpensive fee, from a local Humane Society.
2) I have had feral kittens, as pets. Which I tamed, and they got very domesticated. And I got them neutered. And they were indoor and outdoor cat pets.
3) You also have to, if you decide to keep it as a pet, get it checked for worms or get it de-wormed etc. and checked by a Vet. Then you will know it is healthy or not or has diseases.
4) If YOU take it as a pet to your house, then you need to keep it as in indoor cat. Being there are dogs in your neighborhood. And per Fleas... you simply treat your cat, before you get it all in your home. Go to a pet store to get flea products and shampoos etc.
5) You also need to get a littler box, and cat litter. Cats use litter boxes. I did not have to train mine, to use a litter box... they just used it with no problem. Ask the pet store or Vet, what you will need. Including pet food etc.
You put the litter box, in an out of the way place, where you don't mind having the cat "toilet." And where your kids will not get into it.
Keep it where baby cannot go.
6) yes, some cats shed a lot of fur. Some don't. You "train" the cat not to go on furniture. My cats, did not go on furniture. I trained them. Like you would teach a Toddler. They were good about it. I had a cat "chair" for them to go on. THAT is where, my cats could lay on. They knew that.
7) Yes, get scratching posts. Cats need that.
8) As I said, my pet cats, were initially wild feral cats. I found them at my door, when they were kittens. The Mom had died. I found her dead in the bushes. The kittens, then needed to be cared for, which I did and then they got tame, and domesticated. And became, "pets." They were indoor and outdoor cats. Then when I moved, they became indoor cats and were perfectly fine.
*Your Mom/parents, do NOT seem to want this cat nor to care for it? Thus, you need to either leave it be, or find a home for it, or take it in to your home as a pet.
Cats, are good pets. They are independent and loveable. You don't have to walk them. They use a litter box.
And you train them, not to go on counters or furniture, or beds.
My cats, were behaved.