What did the family do?

1. The family gave special rice cake to their new neighbors.

2. They gave special rice cake to their new neighbors.

3. He or she gave special rice cake to their new neighbors.

4. He gave special rice cake to their new neighbors.

(Which answers are correct ones?)

They're all correct, but it would sound better if you put the article "a" before "special."

. . . a special rice cake . . .