Is it safe to mix calcium carbonate and sodium chloride?
Safe in context of chemistry, physiology or long term health care prevention ?

With zero knowledge, it is safer to stay with commercial products, until you know, what you are doing. 

For mouthwash, calcium carbonate is quite ineffective, being insoluble solid. Better would be solution of sodium bicarbonate ( baking soda ), present in some toothpastes as well.

Salt is not great desinfectant and must be used in really high concentration, that can be very disturbing if swallowed.