What work besides flying planes does an airline pilot regularly do?
There isn't much non-flying duty for airline pilots. 
The most important is recurrent training. Pilots are required to attend a refresher course at least annually. This school covers regulations, aircraft systems, and performance. The performance part involves computing weight and balance, airfield/weather restrictions, fuel required, etc (basically the same thing as a Cessna 172, but with much bigger numbers). Additionally, full flight simulators are used to teach and/or evaluate normal and non-normal (we don't get emergencies anymore) procedure. This training is usually concurrent with the groundschool.
Pilots are required to maintain their approach charts and aircraft manuals; changes come frequently.
A junior First Officer will spend an eternity (at least it seems like it) doing standby duty. This involves being available to take the place of any pilot who calls in sick or is otherwise unavailable.
All pilots must pass a flight physical. The implication is that they must maintain themselves in at least minimal health.

I hope this at least partially answers your question.