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.
Heating doesn't remove water from the air, as you explained. But that's what people assume when they argue that "heating doesn't dry the air" as I've heard in the past. Using relative humidity is a good explanation.