Yes, you certainly can mix warm and cool tones. Although, it is usually done in combination of walls and other elements (furniture, accessories, art.)
Because the walls and trim are on the same visual plane, you should give careful consideration to your color choice so your walls and trim have harmony. Grays (tertiary colors) have a range of undertones so you have a lot to consider.
I suggest hiring a local designer who will asses your space and help you find the right hue. They may also be able to select another color to use in your room that will help bridge the colors in the walls, trim and floor.
An hour consultation is often a vise investment!
Good luck!