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.
Do you not have space for a washing machine either? In many apartments in the US the dryer is stacked on top of the washing machine. It doesn't consume any additional space.
In a smaller kitchen it’s more likely that the dishwasher will be sacrificed than the washing machine (since it can be replaced with a sink)