Have you got things working yet? I believe the power problem is with your inverter, if not then battery which probably has shorted cells. Because when you plug the trailer into electric you hear the inverter fan run for a while then quit it could be the inverter or battery causing it to overheat then quit. Cells in the battery shorting out, just like in a car, will overheat the inverter. The inverter acts as a charger for the battery. If it shuts down everthing will shut down. The heater needs it to ignite the gas, the fridge needs it to run or to ignite the gas to make it run on gas. Furnace needs it for same reason. If you can not ignite the gas then gas opperated appliances will not run because the burners can not light.
Think of it this way. The inverter is a large battery charger. It keeps the battery at 12 volts. The battery then runs everything, lights, hot water heater and furnace. The fridge is the only 110 volt appliance in the trailer unless you have a microwave.
As for the furnace, it is propane but needs the 12 volts from the battery to light the gas pilot. On SOME furnaces you can light the pilot by hand. You can use a match to do this. Your furnace has no vents except the front of the furnace. This is where you will feel the heat come from. This is from the brown box under the fridge. Air will blow out from the bottom right "box" that has the 6 openings in it. This is designed to blow air in all directions. Make sure gas is on and turn the thermpstat up and listen and feel for the forced hot air.
If nothing happens there are a few things you can check. Make sure all breakers are on and all fuses are good on the inverter. next, take the cover off the furnace (the brown one under the fridge). Look for an electric switch and see if it is on, note not all have switches. Next look for a gas valve. It should be where the gas comes into the furnace. All furnaces will have a gas shutoff valve. Some owners never turn on this valve because they never needed heat when they went camping. I found this valve shut off on my fridge when I got my trailer, once I turned it on everything worked fine! Once switch is on and gas turned on the thermostat then should turn on the furnace.
Keep in mind the battery is key. It operates the igniters on both the furnace, hot water heater, lights and fridge. Without it everything runs off the inverter but if it being bad and it causes the inverter to shut down then nothing will work except the gas range burners.
Sorry, I know this is long but I have tried to explain everything. If you need more help let me know.