What can you tell me about my converter/charger?

In those temps you may not make it through the night. If you are noticing things like fan speed dropping off you may be running the battery too low. Many people will recharge and stay above 12 volts or so. Although it's a 12 volt battery your operating voltage range is about 12 to 12.6 or 12.7. When it goes into float it will be about 13.2. You may do well by using a hydrometer to verify your battery condition until you get the hang of things. An hydrometer is the final word.