a bathrom with no extractor fan will be damp and mouldy unless you open the window a lot.
Water vapour from drying clothes is lighter than air and will rise through the house until it escapes or finds a cold surface to condense on.
Some of the black marks (probably condensation) appear to be on sloping celings, which are very difficult to insulate, so likely to be cold and likely to attract condensation. I think they are wettest at the lowest point, which still matches with beads of condensation running down. One of the photos shows pattern staining, with the ceiling badly stained between the joists, but not at the joists, which is typical of a badly insulated ceiling. There's a chance that condensation in the loft is rolling down the tilers felt, but for the moment I'll guess that it is on the inside surface.
You could agitate for the owner to fit a powerful extractor fan in the bathroom. It must vent outside, and not into the loft. When (if) you get one, you can hang wet washing in there, with the fan running, and it will suck the water vapour outside the house. it will also remove the steam from baths and showers.
A modern bathroom fan will run for about 100 hours on 15p worth of electricity so the cost is insignificant.
If your windows have trickle vents, open them.
ventilate the house every morning, for example by opening the bedroom windows after getting up until after the beds have been made. Water vapour will rise to the upstairs rooms. If the windows are misty or wet, open them. If any room is particularly misty (apart from the bathroom) in the morning, there is a chance it may have another source of water, such as plumbing leak.
There are some marks by the skirting but they don't seem severe. Is this upstairs or down? Are the floors concrete under the wood floorng?