When you say none of the "plugs" in the unit had power, are you referring to the 110v receptacles? If so, they will only be live if your trailer is connected to an AC power source (power pole-shore power). The converter changes 110v AC power to 12vdc for your lights only so that you don't run down your battery while plugged into shore power.
Furnace-sounds like the burner never ignited. My furnace will continually try to ignite for example if the LP tank runs out. Generally speaking, the furnace cycle is as follows assuming you have gas and power supplied to it. The thermostat calls for heat. The furnace blower starts and runs for a specified period of time...this is a purge cycle to ventilate the combustion chamber of any residual gas. Next the igniter and pilot gas valve will cycle in an attempt to establish a pilot flame. If it is successful, the pilot will burn for a short period of time then the main gas valve opens and the burner ignites. After the thermostat is satisfied, the main burner and pilot gas valves close. The blower will continue to run until the heat exchanger temperature falls below a set limit and then the blower will shut off.
Sometimes cycling the power to the furnace can help it to reset. Don't just turn the power off and right back on. Leave it off for a minute or two for all the electronic circuitry to "die" then reapply power.
2006 23B Hybrid with 10k round bar WDH
2011 F150 4x4 SCREW Ecoboost, Max Tow, Integrated TBC, 3.73 LS axle, Firestone Ride Rite airbags.