I'd say the unit trumps the manual in any case. Extra or the wrong paper could have been in there, and if you've had previous owners, well you never know what they've done.
When you plug in to shore power, the 120v goes to the van outlets just like an extension cord, so the convertor is not involved in them being powered. Well extension cord with a breaker. If not labeled you can find the breaker for the outlets by flipping them until there is no power in the outlets.
The same 120v goes to the convertor and that is what it uses as it's power source. Again, it should go through a breaker, a different one than the outlet breaker. There might be a third for the fridge or microwave. The breaker being tripped or off will mean the convertor is not powered.
It's quite possible the convertor is not directly wired, but has a plug going to an outlet specifically for the convertor. If it was unplugged, this is another possibility.
Last, the convertor might have a switch. I spent a year with my invertor and never had any clue it had a switch until it wouldn't come on and I looked up a picture online. The previous owner might have turned it off to keep it from running.
Past the convertor there might be a large inline fuse, or auto-reset breaker. Either of these could be bad, but the fan in the convertor would be running. How the 12v goes, which battery it charges, whether there is a isolator involved (sounds like yes) all a different story, but it doesn't sound like that's your problem.
As for your questions, #1. No, it charges the battery and the battery powers the house 12v items. Essentially. It's all one system an a little chicken or the egg, if there is power then it's charging the battery and powering the system, you remove the battery or the 12v from a system with a good battery and you will only see a small voltage drop.
#2. I think this is backwards: alternator charges the starter battery and the isolator, isolator charges the house battery but prevents a charge from being drawn from the starter. The convertor would charge the house battery, along with powering the 12v on the house side, and not charge the starter because A) isolators don't work both ways and B) possibly to prevent overcharging (I don't buy that...)
If it were to charge the starter and not the house, the isolator would pass that charge to the house just like it came from the alternator, so effectively you would be charging both... very unlikely that's the case.
So... your house battery is probably toast. Even if it's not, due to over draining or age, the voltage is probably micro (0.3v). There is probably a threshold voltage after which the convertor won't charge the battery, so that might be your problem as well. The alternator doesn't have any such qualms, so you might run the van, verify there is voltage at the house side and then plug in the convertor (with the van running, with voltage on the house side). I've tricked my inverter into charging like that, but it turned out my battery was bad after all.