Great job for one of your first works. I wish I could have started that good.
As to desaturating your background it all depends on your characters. Your colors are pretty decent already, I don't think a little desaturation will hurt at all though. If your characters/foreground start to blend into the background to the point that they aren't distinguishable then you might have an issue.
I made a partial edit to the image to show how sometimes avoid black outlines can help bring some depth to the image. I messed with the table, sign, file cabinet, and the bulletin board. I changed the outlines to instead edges that are lighter/darker depending on the location of the light source (the sun and the lamp). I did shadows on the bulletin board but none of the other objects I mentioned. (the sign wouldn't need any).
Great job and keep up the good work.
