To begin to sort the problem we need to get some specifics out of the way, so some questions need to be asked first!
Sorry to seem difficult but these things are not simple at times and asking the right questions upfront can get you fixed much quicker if we don't wander around too much!
One big question is which exact RV you have as there are many differences in smaller or larger and to look at drawings of the RV, we have to get down to the right year, make as well as model!
In the 2005 Sunova, there are six models, can you sort down to which you have?
That will allow us to sort down to get the right plan..
But in the meantime some questions may help to make sure we are chasing the right electrical system.
Can you give us more info on what you are seeing/hearing when you say the heater tries but will not run? Do you mean the furnace or another heater like a small electrical heater?
The difference is in the furnace uses 12VDC, while the other small heater likely uses 110AC power from an outlet?? Do you have an AC or DC power issue?
Since the monitor is not any help as it seems to have lost it's "mind" for all things to have failed, I suspect you will need a meter to test the batteries and connections.
It sounds like there is a simple item like a dirty ground or positive battery connection at some point. Good DC battery supply is needed for many things. Things like the monitor, furnace controls, and lights all need good 12VDC.
When you mention doing one electrical thing changes another, it leads me to believe you are only gettting a small amount of battery supply to those, so using more power on one, makes the other not get enought to work correctly.
It "could" be as diffficult as shorts or defects of that sort but it is much more common to have dirty, corroded connections!
Testing the battery voltages at the batteries will be a first step as they are the "foundation" that all the rest is built on.
If no meter is on hand, try starting the RV engine as that may let the engine alternator provide the power needed to help many of the coach things like lights and monitor operateand that will be a definite clue!
It may help or it may not if cables are the problem.