PhD programs and rules at the university vary...some friends of mine said where they received their PhD they were given 100% the same duties and responsibilities and regulations as a full professor. Yet other friends have said where they went they were always reminded they were students.
Some universities have rules about faculty (including graduate students) having a relationship (such as dating) an undergrad, where other universities do not have anything on the books it is just an understood taboo, and others only suggest you do not have the relationship.
Leadership behavior ... I think it depends on the person and again the university. Some use their graduate students to bridge the gap between the undergraduates and the full faculty...some do not. Some graduate students are used to help with advising undergraduates some are not. ABD students are more apt to be used as advisors, or in other more responsible roles since they are almost done with the PhD program.
I believe the best students no matter the university rules or community set up will show leadership, morals, and self - discipline thus being a good example and leading by said example.