Is getting really drunk and dancing going to solve anything?

Solution 1:

No.

You are neither awarded, nor reprimanded in any way for doing either of these. They are, as you said, throwaway interactions. There are a couple of humorous moments to be found by doing so, however.

If you meet Jack in Purgatory and speak with her, you can actually dance with her... After she mocks your lack of finesse on the dance floor.

In addition to Shepard showcasing his dance moves, which are obviously centuries more advanced than anyone else on Earth (except maybe @TimStone from the looks of his comment above), drinking at the bar three times causes Shepard to back away and black out. He wakes up laying on the couch where Aria T'Loak is seated. The two share an awkward glance, and Shepard returns to his cavorting across the galaxy, saving the world, as is his way.