First, WELCOME to RV.net!
Your generator usually starts from the House Battery(ies). Running the road engine will help the House, but if that battery(ies) is/are really low, generator still isn't likely to start.
Are your house batteries Six or Twelve Volt? I'm trusting they're Twelve because if they're Six and only one hooked up, nothing would probably work.
Your furnace may burn propane to produce heat, but it has a Control Board that is 12VDC and runs off your House Battery! Same for your Fridge and probably your Water Heater.
I think you probably have House Battery and/or Connection problems.
But ALWAYS ALWAYS ALWAYS post the Make/Model Number of things like Appliances, Generator, Chassis, etc.
If God's Your Co-Pilot Move Over, jd
2003 Jayco Escapade 31A on 2002 Ford E450 V10 4R100 218" WB