I have never covered mine and this article also seems to recommend not covering it:
Your central air conditioning unit
consists of a compressor and
condensing unit placed outdoors in a
metal housing. These units, built to
resist the weather, generally do not
need a cover. In fact, covers can
cause problems because they trap
moisture and create an inviting winter
home for small animals.
Professionals who service the units
tell me that most of the damage they
see in spring was caused by rodents
living in the units and chewing on
wiring.
If your air conditioner is subject to
falling ice or other debris, you could
cover its top with a piece of plywood,
plastic or metal held in place by a
weight.
Reference: http://misterfix-it.com/blog/?page_id=36#cac
They make a great point that a cover would provide a perfect winter home for animals which would cause damage to your air conditioner.
My guesses without more information would include the crankcase being overfilled with oil, a blown head gasket, or damage to the piston, rings or cylinders.
I doubt that you have the generator running while it sits at an angle, which could let oil slop over into the carburetor float bowl. Although that's a possibility, too, if you have it sitting on a slope.
Since the engine apparently always produced a little while smoke, and since the smoke doesn't start for a while, then doesn't stop for a while after shutting the engine off, it's probably heat-related in one way or another. As the engine heats up, it expands. If there's a damaged head gasket or something like a hairline crack in a cylinder wall, it could open up just enough to let oil seep through once the engine is hot. Or if you have something like damaged or ill-fitting piston rings, oil might start squeezing past the rings once the engine heats up and the cylinder expands.
You mentioned that if you hold your hand in the white exhaust, little oil droplets condense on your hand. The same thing is happening inside the exhaust manifold/pipe, and the heat will cause that condensate to keep smoking until it is all burned off or the temperature drops low enough for it to stop smoking. If the exhaust valve is open at all after the engine shuts off, any oil in the cylinder will burn off and push white smoke out the exhaust, as well. Especially if it's some kind of hairline crack letting oil into the cylinder, which then closes back up as the engine cools.
Of those possibilities, aside from overfill or running the engine on a slope, the head gasket would be the easiest thing for you to fix yourself. You would need a torque wrench and specifications on the correct bolt tightening pattern and torque for the head bolts for your specific engine.
Or it could be something completely different.
Here's a potentially handy page from Briggs and Stratton, but it might be time to call a local small engine repair shop.
https://www.briggsandstratton.com/us/en/support/faqs/engine-emits-white-blue-smoke
Best Answer
Ideally the manufacturer has engineers that put a lot of thought into this, and the manual spells out all the recommendations for even the most extreme operating conditions. But, lots of times you don't get that, it just says "use 10W40" and that's it.
If I think my operating conditions are likely to be way out of what the manufacturer's recommendations account for, I might deviate from those recommendations. I might use a more viscous oil to deal with the heat, or less viscous oil to deal with the cold. Your conditions in Chicago are typical for huge parts of the country, I bet the manufacturer's recommendations are fine for Chicago. I'd probably leave well enough alone and stick to the recommendation.
If the engine has a lot of miles / hours, and seems to have significant wear, I do switch to something a little more viscous. If a car is burning more than say a quart between oil changes I will suspect it's worn. I figure an engine with some wear is a little looser and will blow by a little more oil (past the pistons into the combustion chamber) so a more viscous oil might make sense, like switching from 10w30 to 20W50, especially in the summer months. I have seen this make a measureable difference many times with high mileage cars.
I haven't really seen the extra viscosity make much of a difference in starting in cold weather. I think the drag you get from viscous oil is pretty minor, better to focus on a good quality battery with adequate cold cranking amps and good cold weather performance if you're concerned about hard starts in cold weather.