Yes, the evaporating cools down the room, though I can't say I ever noticed it. In terms of energy efficiency compared to a dryer, it's much better in summer, and not much better in winter, though you get a humidifier as a freeby.
Yes, it's a chore, a full load takes about ten minutes to hang, and it's real boring work.
How does adding latent heat reduce the sensible heat? Are the clothes washed in very cold water?
It relies on the air not being saturated (and for real effectiveness, requires relatively low humidity). It's quite popular in dry climates in the form of a "swamp cooler".