You may decide on different pillows one of these days so you don't necessarily want to use the pillow colors for the draperies. How about a color from the rug, in a solid, or a simple stripe? The rug, being more expensive than pillows, is less likely to be changed soon.

I suggest moving the media console 3" to the left so that draperies over the big window have space to hang on both sides of the window, using 1" metal rods hung just below the ceiling. I would add 2" wood blinds to the window to avoid the expense of custom draperies. Or you could use bamboo shades there. I prefer wood blinds for the sun control they offer. Bamboo shades are quite charming, but they're either up or they are down.

On the doors, you could have roman shades in the same fabric, or just wood blinds.

The rug has chartreuse, medium and dark grey, light and dark blues, and a hint burnt orange in a kind of abstract wave pattern.

an idea for drapes, but I wouldn't do more than white sheer cafe curtains on the french doors:

Home Depot sells a easy to install blind insert for French doors. They look built in. Added bonus they won't bang around because they are incased in the glass. You can raise and lower them completely or use them like a blind. Clean the glass like you already are. It's a good option for controlling sunlight. You could then hang sheer curtains for the look you want. I have installed these inserts on two sets of French doors and on two other half glass doors and absolutely love them.

Pics of inserts

I love mullioned glass. I just hate covering them.