1. How do we get the furnace itself to turn on with the blower?
We had the same problem and it was caused by low voltage.
The furnace is designed to operate at voltages between 10.5 to 13.5 VDC.
When the voltage from the batteries gets too low, the furnace fan will start but the burner does not light, so the furnace just blows cold air. Not too helpful.
The reason for the furnace not lighting is something called a sail switch.
The sail switch is an on/off device. (normally in the off position) It gets it's name from the "sail" or paddle that is attached to the switch mechanism. As the blower comes up to speed, it blows air onto the sail with enough force to push the switch closed, thus allowing electrical current to flow.
So with low voltage, the fan never blows hard enough to close this switch and turn on the gas for heat. The race is canceled because there is not enough air for the sail.
Our voltage drop was caused by the buss strip in the house battery compartment. The buss strip has four circuit breakers, a 50 amp for 12 volt
distribution panel, three 30 amp breakers, one for the steps, one for the dock lights and one unused.
I found the first breaker (50 amp) that supplies the power to the 12 volt
distribution panel was loose. There was evidence of arching and the heat had melted the plastic assembly that holds the buss bar and the breakers.
The breaker assembly must be something Airstream had made special. None of the auto parts stores had ever seen anything like it, and neither had the local RV dealer. I fixed the part myself.
I cleaned it up, used some super glue to reattach the broken plastic and it worked just fine.
Here is a link to a furnace troubleshooting article. This site has other info you will probably find useful as a new owner.
Troubleshooting the RV DSI Furnace
2. Is there a way to increase the volume of air blowing out of the outlets (there are two outlets we've found that air comes out of when the blower turn on - one is from an outlet that is at the end of a flexible duct under the jack knife sofa, and the second is in the wall in the back bedroom.) The one in the front under the sofa has hardly a trickle of air coming out. Is there a filter we're supposed to replace somewhere?
There is no filter. I found some holes in the ducting here and there and patched it with Red Green's secrete weapon, Duct Tape. Fixing the holes helped, but don't expect the same flow you have closer to the furnace like in the bathroom.
3. There is another outlet in the floor near the exit door, but no air comes from there unless I turn on the rear heat switch and then only when the engine is running. Is this a separate system and tied directly to the engine heat?