Question

Kathleen and (your name here) both like to go out with you and Mike. The name in bold should be replaced with he. himself. itself. him.

Answer 1

Its either He, or Himself, but its more likely to be Himself.

Answer 2

which is the name in bold? Mike?
In which case, it should be:
Kathleen and (your name here) both like to go out with you and him.