The wood paneling appears to be quality, if so, it deserves a Pro cleaning. After having it cleaned, I would suggest a porcelain travertine lookalike, in a warm shade to complement the paneling. Paint the cabinet if need be...not the paneling. Porcelain is a snap to keep clean...vacuum and use a cheap floor steamer. I have two dogs, 200 pounds and 125 pounds, so I speak from experience.

Then I would say you have the two PERFECT floors for your situation. Tile in the "wet" areas and carpets in the rest. Tile doesn't really care about humidity control. Nor does carpet. To get a rigid floor in here (laminate, cork, hardwood, engineered hardwood, vinyl, etc) you might find the lack of humidity is going to be the defining factor. Any floor you put in there will lose warranty the first day of install. If you are OK with that, then go for it... I would personally install a hardwood (engineered in a narrow plank would be better, but it will not rid you of the problems of movement) in the "shoulder season". That means spring or fall. I used to live in the Okanagan Valley in BC (Canada) just north of WA state (Omak WA). The high desert = same concept. The summers have humidity around 15% until a storm rolled in = 90% humidity. And then it would drop again with forest fire season. And winter was then crisp and dry. Pellet stoves will dry out the air nice and fast. So the 20% humidity outside will drop even lower as soon as the stove is fired up. For any rigid floor (other than tile) these humidity levels are far too low. They will shrink and gap over the years...the gaps getting wider and wider as the floor ages. So long as you are OK with this and will love the wood for what it is, then go for it. But buyer beware of your home's limitations. And let me tell you it gets HOT in summer! My friends and family live through 40 C (104 F) summers every year. This is climate change at it's best. Thirty years ago, the hottest summers would be 38 C (100 F) but that would only be every few years. The average summer = 35 C (95 F). Those cathedral ceilings will help, but they won't get rid of all the heat. If your sleeping quarters are "upstairs" you will want to find a way to get the heat out. Just a little FYI from someone who has lived in a similar climate.

The floors are stronger than the couch So bringing some of that carmelish looking color up like in a chair as this photo shows They pull the grey couch up on the walls With a greig color that has a warm undertone I think that wall color would work on all walls But this real designer uses a darker greig In the adjoining room I love the mirrors

I would paint the doors. Any color. Without seeing the entire room, or see the colors you are working with throughout the home, I won't suggest a color. I would Beef up the trim moulding around the door(s) keep it white as the featured trim in the room. With the highlighting of the trim/wainscotting, I'd paint the walls a hint of color to show off the features. some fabric on the windows for texture? I would leave the bar the natural wood tone you have going. shelving in the alcove, nice... I would personally go with a black or Ebony shelf, but same... I don't know the look you're going for. Keep the woods in the same family, Is the floor carpet, or an area rug? I'd work on the flooring, Bars and carpet don't go.