Are you saying light comes from the north until late in the day, or are you saying there is a window facing north and a window facing west.? The reason I ask is that light that enters from a north-facing window is bluer and lacks intensity. Subtle color is washed out to gray. The Inchyra Blue is very dark, looking more charcoal with a green tone to me. I would choose the Cooks Blue. Here is a comparison I put together of those two colors with some blues from your rug.While the actual colors may be off due to how color can vary from monitor to monitor Cooks Blue is brighter and won't fade as much in northern light. Also, having a brighter blue on the walls should bring out the blue in your rug. The only way to know how it looks in your room is to buy a sample and paint a board as you did for other colors. Keep in mind that the color temperature of light bulbs you use will also affect the colors of your walls and furnishings. Color Temperature is expressed as Kelvin but is not a measure of heat. Warm-white bulbs have a color temp of 2700K which is kind to skin tones but gives a yellow light. Daylight bulbs have color temperature of 5000K to 6500K, but are harsh to skin tones and emit a blue-white light. Both CFL and LED bulbs come in various color temperatures. I recommend you choose bulbs with color temperature between 3500K and 4500K for a more neutral result. HERE is a photo of how different K-value lighting affects the appearance of color. You might have to visit a lighting store or light bulb specialty store to find neutral white bulbs. As to ceiling colors, people seldom look up. Unless your light fixtures are dramatic or the ceiling color is distinct I doubt anyone will notice if two adjacent rooms have different ceiling colors as long as there is a Framed Passageway between them.

I'd keep it neutral(beige, off white). It also depends upon what else you're going to do in the space. Some good choices: https://www.clubfurniture.com/collections/apartment-size-furniture

It's really hard to give advice without pictures...the safest options would be to match the tiles or use carpet again.

Yes, arrangement looks much better!! If you are anxious to paint, you could roll out the Natural Wicker, as you are going to need a primer anyway over that deep paint. Won't hurt to use is as basically a primer. I can't really recommend a paint until you are finished with the room. Your next step will be chairs, lamps, end tables, lamps (I would get a nesting coffee table or oval), art and drapes (which are too short).