If you've got a child and you are living in a house with them, and then you buy a new house, how do you get the child to move to the new home with you? I don't have a spouse or anything.

You need a bed and dresser.

Don't know when there is no spouse, but I would think there should still be a dialog option when speaking to the kids? Can someone else answer this for sure?

You get dialogue options with each child for moving, I never risked complications and spoke to both children and told them the same thing. One of my characters has yet to get married.