Hot dogs that you buy in the store really are already fully cooked and safe to eat right out of the package but heating them up makes them taste better and makes for a much better lunch, dinner etc.
I used to eat cold hot dogs as a kid from the package when I went shopping with my parents and I would reach in the back of the shopping cart and open a pack of hot dogs when my mom wasn't looking and I was 4 years old.
I would eat the hot dog while we were shopping and finally she noticed so she put them further away out of might sight after that or I would eat the hot dogs before we even got home.
I never got sick from eating the cold hot dogs since they are cooked already.
But for a food establishment the health department will require a minimum holding temperature of the hot dogs to be around 140 F and if the temperature drops below 140 F then they are considered unsafe to eat by the health department.
Also according to the health department and when I worked for a restaurant the health department stated that we must heat the already cooked hot dogs to 165 degrees F and then keep them held above 140 degrees F until served.
Running a restaurant is a good business but you must deal with the health department a lot and go by their guidelines to make sure your customers are safe.