a cool gray

If you paint your woodwork (minus the french doors) a light creamy color, I'd go for a light grayed rosy-purple color. It will be "girly" but still sleek. It also won't compete with the warmth of the wood tones of the floor and rug. I have a gray and yellow room posted on my page - but you'll notice the floors are light and do not have red undertones as yours appears to. Think a contemporary blush tone with a tinge of gray to it.
Check these out - Benjamin Moore: Tropical Sand, Pale Cherry Blossom, Charmeuse, Soft Satin

Great chairs in the other room, by the way!!

I would paint the walls the same golden yellow as the rug.

I would paint the walls in the same cream color found in the rug. Let the rug make the statement and allow the walls and wood to enhance not fight with the rug. It will make the room look fresh.

What if we're painting the woodwork white? Do I need to ditch the rug? I want something very bright.

If you are going to paint the woodwork white then paint the wall above it a much lighter shade of yellow than the rug shown in the picture. Find the same color yellow as in the rug on a Benjamin Moore color strip and go approx 3 tones lighter. This will co-ordinate the room colors without making it look like you need sunglasses when entering. I recommend painting the woodwork in a satin finish, using "cloud white" (Benjamin Moore) and paint the doors as well.

we have some concerns painting the doors white because they open up to rooms on either side where it might look strange. do you think that will look terrible?

It will look odd having all the woodwork white except the door. It will not look odd when the doors are open into the adjoining room because it will relate to the "white" room.

I would avoid painting the walls yellow. In my opinion it would be a lot of the same. I think painting the french doors would be fine, but don't think it would look bad if you didn't. You are making them a statement. Purposeful design works.
Definitely bring the white paint chips in to the room- Some whites that work for others do not work for everyone! Look for a warm white.

Thank you WOODMAN! You've been super helpful.