I would wash any Najis area after interaction and would ensure that there is no nijasa or hair during prayer. I follow Sayed Sistani.
Yes, but it is better if it can stay outside (such as in a dog house, if that is possible on your property; these days, they do have outdoor heaters for animals if that is a possibility).
I am sure you are familiar with the hadith saying that having a dog inside affects acceptance of prayers; Allah knows best. Of course, we are also supposed to be compassionate, and it is not compassionate to let it freeze to death!
Yes you can interact with it and train it.
Yes, be sure to observe taharah for salat and any other necessary things such as dishes.
People have had dogs as work animals for centuries, and even the Ashab al-Kahf had a dog around them. As someone once said, it is not the dog's fault that it is najis!