Typically the chassis batteries would not be charged by the converter from the factory. You can put in some sort of an echo charger (one popular, good, and inexpensively priced unit is the LSL Trik-L-Start, or its larger brother the Amp-L-Start), or a small standalone (smart) battery charger dedicated to it. If you aren't planning on driving, which sounds like it may be the case, you could also remove the chassis battery (taking care that the positive lead doesn't ground out against anything) and jumper across the emergency start/isolation solenoid and run everything off the house battery.
It's hard to know if the converter is properly charging the house battery without making some measurements, particularly the voltage it's putting out. Before suspecting it of not working properly, though, I'd recommend verifying that the battery is good and the connections to it are solid. The generator starter is probably the largest load that it sees; cranking the generator takes somewhere around 100A on my motorhome. It doesn't take much of a poor connection at the battery or the generator to cause a significant voltage drop, significant enough to make it hard or impossible to start the generator.
(If the generator has trouble cranking even with a good battery, it's possible too that its starter motor or solenoid or the internal connections between them need help, too.)
The extra vents, at least some of them, could possibly be air return vents for the furnace.