Re the car, I THOUGHT you were going to ask the oft-asked question here that "should I let wife take the car when she's with the kids, because she's also using it to go be with OM," but this is when YOU have the kids???
Not even a difficult decision, in my mind: HELL NO. I would not do ANYTHING that enables her affair, unless there's some overriding concern with the kids.
In whose name is the car titled? Who paid/pays for it?