"He is playing in the street" is correct. "In the street" indicates that he is playing within the boundaries of the road, where the cars drive. "On the street" refers to the area alongside the road.
If Fourth Street is the name of the avenue, then the proper phrasing would be "They live on Fourth Street" (without "the). If you are specifying their street in comparison to something else, then you'd say "They live on
the fourth street past the school."
Fifth Avenue is the name of a street, so you'd say "I went shopping on Fifth Avenue." (no "the")