"Work it" usually means, "strut your stuff," show off your assets, make some physical attribute (figure, walk, style of dress, etc.) to "work" for you to make you attractive.
While not proven or confirmed, the expression may have originated among "working girls," i.e. prostitutes/street walkers, who would make a sultry walk etc. "work" for them to attract business. If it did have that connotation, it has been lost with time, and now usually is applied more innocently.

when you're hungover as shit after a spending a night drunk as a skunk

-Make use of something you have to obtain something. Can be of a monetary value, physical attributes, or status.

-Action to persue a goal.
(Present tense: Working it)

(verb) means:
1-to show people how attractive and talented you are.
2-to solve the problem.

To put something into action or to give it the "old college try"

Means to jackoff.

(noun)
1. Vocoder-intensive song by Daft Punk.
2. Flash animation made to the aforementioned Daft Punk song, by evilzug.com.