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?
You're not adding latent heat to a room, you're adding water, which when evaporating uses up the heat energy in the room which cools it down.
Look into how swamp chillers work.