Outside 24/7/365
Plugged in convertor ON, batteries connected (this is provided you have at least a 2 stage or 3stage modern converter which will not boil the batteries to death).
NO HEAT what so ever. Unless you are planning to camp in the winter, winterizing the water system IS your best bet.
I personally would not recommend depending on a small heater or even using the RV heater when not in use. Just wastes your money on electric and propane and if depending on it to keep your water lines from freezing you might find that you will need to keep the temps up much higher than 40 and if you have a power outage risk of freezing..
Keeping battery charged..
Provided you have a modern 2 or 3 stage converter you can typically leave the converter on with battery attached. Just be aware than if you have a power outage (or somehow the RV loses power) there will be a constant drain on the battery (from the 12V devices in the trailer like fridge control board and propane detector) which can flatten a battery in a week or two.
Alternately you can simply disconnect the battery and once every month connect the battery and power the converter a few days to top off the battery..
Or you could pull the battery and store in your garage, but you still will need to put a charger on the battery at least once every three months to top off the battery..
Fully charged batteries will not freeze however, partially charged or completely flat batteries can freeze. It is always best to ensure the battery is fully charged at all times for best life of the battery..
Leaving a battery in a partially charged state will reduce the over all life of the battery..