They are called babies or infants while in the nest. Then for their first year they're referred to as Juveniles. After their first year, they become adults and are simply called squirrels. No, the male squirrel leaves after the mating. The female raises the babies by herself.
Urban squirrels tend to have a smaller territory, they sometimes have to share resources to survive.
They also have learned, to coexist with humans. While rural squirrels are more protective of their territory, and they have more natural predators to deal with. Squirrels claim an area, usually one to seven acres in size. They will build their nest in this space. They will find, and store their food in this area.