What do the Lethargians do in the Doldrums?

The simple answer is: not very much. The Lethargians, as their name implies, are lethargic, or lazy. These slimy little creatures live in the Doldrums, a swamp of laziness, ignorance, and dread. Here they spend most of their time sleeping, and during those relatively few moments when they're actually awake, they follow ridiculous laws such as not thinking or laughing aloud and only smiling on alternate Thursdays. They also have laws against reasoning, meditation, presumption, and speculation as these are considered unethical.
One might think that the Lethargians, with their chronic inertia, wouldn't have either the time or the inclination to breed. But fortunately for them—and unfortunately for everyone else—they can simply divide and multiply themselves like amoeba.
Nevertheless, for all their numerous faults, the Lethargians do serve a useful function in the story, as they appear as a warning to Milo of the dangers of a life of boredom and inactivity.