>Has anyone every tried using a Submersible Aquarium Heater to warm sugar water inside a hive feeder?
I haven't nor know of anyone who has.
>Would the bees take the warm sugar water even on cold winter days?
If it's close to the cluster, yes. If it's far from the cluster, no. But it would certainly help on those warm days and cold nights where the syrup never warms up enough during the day. But it would also cause the syrup to evaporate causing moisture in the hive.
>Would the Aquarium Heater located in the sugar water also help to warm the hive?
Probably. But the humidity wouldn't be a good thing. Your climate may affect how well or how poorly this works.