My thinking on the shadows on the ceiling below the bath are old leaks that were not primed properly and the stain is leaching through the paint job.
You could get some good primer and paint a box over each shadow and then paint a box over the primer box with finish paint. I am afraid that the patch of new paint will show and be worse than the shadow.
Your basement problem I am not able to offer suggestions. You might just need ventilation and possibly basement windows.