I painted my daughter room red. I primed the walls, used two coats of red eggshell and some areas look darker than others. Especially around ceiling, trim and corners where trimming is needed.
Yes, a flat paint can be applied right over the satin. Make sure to use the same type of paint, example- latex over latex.
The dark areas can be caused by the slight differences between brushing and rolling. This can occur with any dark or bright paint that has a lot of pigment. Using the best paint that you can afford will minimize this effect.