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.
Definitely not possible in the flat we live in during the winter unless we run a dehumidifier.
You are right about absolute/relative humidity, but our basement feels damper in winter than summer.
You can get a related effect with thermostats which raise and lower the house’s temperature over the day as air expands with increased temperature.