Requirements for the Slug Club?

I'd say 16, after OWLs. The ones Slughorn noticed during classes and who did really well at their OWLs would be invited to join.

I don't think there is an age. Ginny was there (she was fifteen), Harry was there (he was sixteen), and Cormac was there (he was seventeen).

Slughorn chose people who would become powerful (in some fashion) later in life. He wanted to make connections. If he saw someone was powerful (or famous) and going places, they were in.