Woven wood blinds. And Bali makes a woven wood slider with three to five panels for sliding glass doors. Ours is a three panel. Drapes on sliders get dirty fast.

I would do some blinds on the bottom and leave the arch since this is a play room make sure the blinds have no cords.

Like KR said, try to get something ordered before you move in. We moved 18 months ago and have been gradually replacing all of the window coverings in the house we moved into. COVID has caused materials shortages and for some types of treatments you may have a long wait. The answer on which type to use depends on what style you are decorating in. While the solar shades sound great, I don't know whether they would look right in a traditional home. They wouldn't be my choice. Some people like the look of interior shutters. Since you're doing a whole house you should be able to find a retailer who would be happy to spend hours with you to look at options. You can ask them what they sell the most of. If you aren't getting windows with a UV coating (we did that at our previous house when we replaced the windows), have a film applied to the inside of at least your south-facing windows if you don't go with the solar shades. I use a lot of 2 inch vertical wood blinds that can be angled to keep the sun out. Patricia is right, the sun will discolor wood floors and fade upholstery in pretty short order.

If you want curtains, 4" above window is plenty of room to mount a rod. If you wanted them higher, there's really nothing you can do about that. Not sure what you mean by trouser length, since they come in different lengths. Just touching the floor or 1/4" off the floor is good. Can't see the window to the left, but if you have the same space as the right window, then you should have plenty of room to stack.