Furnace is a 12 volt device, if the batteries are low it will not work. I have a trouble shooting chart I'll append to this message.
Fridge should indeed get power from the generator, if it's not the Auto Transfer Switch is suspect. However that said, some generators might make power the fridge is not happy with.
This document was created while I was diagnosing MY furnace.. I am told I missed one possible issue... Sadly I forget if I updated the document or not and the issue Bad connections anywhere along the line can lower voltage
When the T-Stat calls for heat the control board starts the blower (1)
The blower blows closing the sail switch (2)
The control board then opens the gas valve (3) and begins sparking (4.1)
Gas Flows (5) And ignitets (4.2)
The flame heats the flame sensor (5)
The flame sensor sends .480 volts to the control board (6)
which then detects the .480 and continues operation (*)
When the T-stat says "Enough already" (RV is now warm)
The control board shuts off the gas flow (*)
and then turns off the blower after a cool down (*)
*: At nearly every step the control board is involved. IF IT IS BAD, things will not work.. I recommend replacement boards from Dinosaur boards.. When Mine went out the Dino board was less than half the price and when opened the box the quality of construction and workmanship, and then design was very clear to my trained eyes.
1: No 12 volt to furnace, blower motor shot
2: Low 12 volt, air ducts clogged, blower "obstructed" bad sail switch
3: Bad solenoid, Clogged Gas Jet
4: .1: Bad ignition circuit (high voltage) on control board. .2: Points too wide (Very common it seems)
5: These do not often fail but they can
5: Bad connection, NOTE:
5 and * Bad wire.
* applies to all steps by the way So does bad wire/connection.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times