Modern Generac household enclosures I've been next to seem pretty quiet, but there are lots of variables......Not enclosed at all right now, other than the factory enclosure. I put the 4 x 4 posts in as sacrificial bumpers for the tractor, mowers, etc. The two at the house end are still around 6', so the thought is some 5' dog ear boards as a sound baffle. No air flow issues at all.
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Window A/C tend to leak sound and air, so improving that may be enough. I'd be tempted to build an "acoustic" (whole field of engineering right there, but as per the vid I posted, even crude changes can yield significant results) wall parallel to the house face, next to the generator. That ^ might be plenty on it's own - in my neck/woods, I'd also be tempted to have a sloped roof from the new wall to the existing rear posts.... but that's probably more because of the snow/ice here..... may be little/none your way.....
Wall one-side + roof, but still open on 3 sides shouldn't create any heat issues......
Rgds, D.