I can't inspect your jobsite, but from your description I can venture an educated guess: Your house is built on a concrete slab and that concrete is emitting moisture. That moisture is condensing on the vapor barrier and is resulting in the liquid water you see and, consequently, the mold and mildew growth. Before you install any new flooring product, find a flooring specialist who is knowledgeable about moisture and concrete and who can give you a prescription for correcting the problem. Lots of ways to reduce moisture transmission through concrete...the least costly being to correct any standing water issues outside the house near the foundation by providing roof runoff pathways to lead water away from the foundation and even by removing plantings that may be contributing to the problem. There are topically applied moisture barriers in both liquid applied and sheet versions that we use to 'seal' concrete slabs and prevent moisture migration.

Do you think chair rail would be a bad idea in the living room and hall but not in the kitchen? Or should I stick with all solid walls no chair rail

I have stained trim in my entire house but not one piece of my crown is stained. That would stick out like a sore thumb in rooms with only 8' ceilings. Racing stripes is the term used in rooms that have all that horizontal stained wood. A picture would be helpful. Whatever you paint, make sure your crown and chair rail are the same color. Are your windows painted or stained? I would assume stained. If you keep the baseboards and door trim stained, then the windows are fine to stay stained. However, you may decide to paint the stained trim once you see the clean painted trim, and if you do, your windows would also need to be painted.

I think that a good hefty crown molding partway up in the atrium area (say to the base of the second floor) might be a good idea to give the eye a break as it goes up the walls. It would also help you however you when you are differentiating one area from another. Especially if you paint above the crown molding in another color, perhaps to match the upstairs hallway area. Rather then molding to differentiate areas you might want to consider paint. different colors for different 'rooms' to break the visual space up. Keeping it all the same color makes it *look* all connected. Say a very light yellow tint in a dimmer 'room', maybe a blue shade or tint in a room that should be more relaxed, etcetera. You could also use your furniture to provide separations so that one area dosen't blend into the next. Rugs (they can be rather inexpensive too) can define a space very well also. As an example: In your living area you could swap the hard wooden chairs for the sofa, put the softer chairs where the sofa was with the round glass top table between them, slide the small table with the glass bowl deeper into the bay window, and place the hard wooden chairs facing the soft chairs and tucked back against that curved staircase. (my thinking being you'll play up the bay window and have a nice backdrop for the bowl, the curved stairs will set off the curve in the wood chairs, and the couch will provide a visual break from the next room.) You could pick out a color you especially like in the couch or rug and paint the walls there that color or a similar shade. From everything I've been reading molding is excellent for setting a sense of formality. I don't know that it would have enough weight to really set one area as different then another though. Hope this helps. I know it isn't quite what you were asking for.