The style can be different:(example) panels to the floor on the French doors and Roman shades on the windows, but I suggest keeping the fabric the same.

You could put up a double curtain rod. Hide the short heavy blackout curtains behind a long pair and pull them out when you want privacy. The long pair can be more decorative. It's done often in hotel rooms that have under the window units.

I had a similar window in my house. I just ran the curtain rod right up to wall. I put 1 or 2 of the curtain rings on the wall side of the pole support, so the curtains sort of hide the support and stay over on the side. Nowadays I prefer the type of rod with no finial (or just a cap on the end of the rod). Just keep in mind that the curtain will cover a little bit of the window on the right. But honestly, I don't think anyone would ever notice. There are also this type of curtain rod I've seen a lot lately: On Ballard Designs, they call them French return. I think they work really well for blackout effect, which might be nice in a nursery. Here's a blog where someone did something similar, though in thinking about it, I'd probably prefer the "French return" style. https://hicks-house.com/2014/04/17/drapery-dreams/

I don't like the wainscoting in the music room. It is too phony looking, as if it was an afterthought. Can you remove it? Yes, I like painting the music room all one color. Have you considered any other color for the music room, working off the dining room color? This is B.M. Sag Harbor Gray. It appeals to me because it is more of a mellow green/taupe.