Running, and warm enough so the coolant is cirulating, is good, but not entirely needed. ALWAYS BE CAREFUL the system isn't hot when you open it, or you can get burnt. Always check the amount of coolant after the car has been run and warmed up a bit and the thermostat has opened, which allows the coolant in the tank to then move through the sytsem. If there were empty spots before the liquids started moving, the amount in the system may now be low. That depends. Is the engine hot? If so, yes, it should be running. If it's already cooled down it doesn't matter. On the other hand, if it's REALLY hot, you should wait for it to cool before adding coolant.