As far as color recommendation goes, I'm at a loss. What's in the adjacent area? I'd be inclined to stay light, maybe even white. Color choice is so subjective that if you ask 100 people you'll probably get 100 different answers.

Looks like a nice, spacious closet! You could add a narrow, marble-topped console table with creamy colored legs. Go with a light colored marble top. This would bring some lightness to the space and add a new dimension to the wood of the floor and walls. And I've seen pics of dreamy closets with something like this and people may place accessories or other things on it. You could also hang a mirror on the walls.

That pic looks just like yours, with the cedar walls and wood floor. Love it when I can find a pic of something similar because it is so hard for me to envision possibilities without actually seeing it. I'm not sure if you need to stain the floors just yet. Perhaps live with it for 6 months and then evaluate. In the pic, there is a rug, too, and that looks pretty.

I've read people mention lightbulbs in other posts as a way to brighten a space. I think the daylight bulbs are the brightest so those will help too.

Another thought...on the shelves you could add a couple of cream colored canvas bins to break up the wood. Once the clothes are in the closet, maybe the cedar walls won't seem so "sauna-like."

Have you decided how your going to hang your clothes? With that layout, you may be able to put in-season in the front part or most readily accessible area and out-of-season in the back.

I'd make sure all the hanging system pieces are white or light metal. Maybe a tufted stool or bench, something clearly not belonging in a sauna/spa. I might have kept the carpet, but second best is to put a light natural finish on the wood, with a matte finish, to keep it as light as possible and out of the visual space.