I absolutely love heading down to the southern part of Utah in October. It varies of course but the days tend to be quite pleasant, the nights cool to cold. Great hiking and exploring weather, and for us, we would much rather have to use extra blankets or the furnace at night than roast in the heat.
Regarding the possibility of snow, it's possible but unlikely, and should you encounter any it will more than likely be a short lived event with little to know impact for the major roads. If you're lucky enough to be able to have flexibility in your travel plans, even Dec thru Feb are great in the Southern national parks as the snow adds a lot of beauty to the landscape. Even in the dead of winter the roads are generally cleared up within a day or so of a snowstorm.