How to control it so that is begins with denser mist in lower scene area and slowly gradient-wise continues to the upper area?
Is it possible to restrict it to only some part of the scene?
By upper/lower, do you mean in world space along the up axis? If so, you can use the method I posted about here: https://blenderartists.org/forum/showthread.php?401768-Gradient-mist-with-z-pass-in-compositor-when-camera-moves&p=3067705&viewfull=1#post3067705
If by upper/lower you meant background/foreground, just run the mist pass through a color ramp node and remap it as you like.