"Next door" metaphorically - in the same area.

I'm assuming it's not rented because of the type of house it is, and also because of Mr Dursely's income. The only way that kind of house would be rented would be if they were there very short term, or students - which they're not - or if it were social housing, which the Durselys wouldn't be living in. They'd think they were failures if they didn't own (apart from the mortgage, of course) their own homes.