Thanks for your concern, gents. The garage is a completely separate unit -- about 30 feet from the house with nothing but air and a sidewalk connecting them (well, wires and pipes, but they're not very good carriers of noxious gases. Yes, I could build a dog house - that's a fine idea - but since the generator is in the garage and separated from the house anyway -- I don't think it's necessary. I'm always (repeat ALWAYS) cautious when going into the garage and the generator is running - open a door first to make sure there's a fresh air supply just in case the unit has built up some stinky stuff in the garage. If nothing else I like having the generator in the garage for the extra heat it provides. The garage is already heated and very well insulated (6" walls, 12" ceiling), but what the heck - more heat, for free, is fine. The insulation also helps keep the generator noise down to sleeping quietly levels, too.
As for starting it more easily - well, no matter how well the (outdoor) doghouse is insulated - it'll almost certainly be at ambient outdoor temp when I go to start it (unless I provided a heat source in the doghouse). The garage thermostat is set at no lower than 41F so the generator has that temp to start. It usually starts on the second tug of the starter rope. I do run it every few months to make sure -- let it run for 5 - 10 minutes to make sure the thing has had a good warm-up. As for gas -- I keep a 5-gallon jug of it, clearly marked "For Generator Only" right near the generator. I put Stabil in the gas so it'll easily last a full year, and then I do run that gas through the Subaru engine and buy a fresh 5-gallon jug.
Here's a photo taken from the house. You can see the garage on the far right side, and you'll have to trust me when I tell you that it is completely separate. I just don't have a photo handy that shows it