Here is the whole house view, but with the old door.

the portion between the door and sidelights is a white, nonstainable wood

I'd paint these brown trying to match the stain, and darker is better than lighter.

little piece of trim that’s unfinished nailed to the top

Stain these to match the door.

Dark brown or black. Do the white trim around the whole unit the same color.

Hi there! You can paint the part between the door and sidelights white and also paint sidelights white, it will less emphasize that part. But I think door and sidelights look amazing now, and it would be good if you can pick up the right color for that part to match the door, like wenge and leave door as it is.