I would definitely check the fuel. Fuel from the pumps can deteriorate within 30 days. Even a fuel stabilizer will only "slow" the deterioration. With this in mind, your first step should be replacing the fuel.
I would not be concerned (as far as this incident goes) with the air filter cover and base being damaged as long as there is no visible damage. Long term, you might have a weakened base/cover since carb cleaner and those types of products will weaken plastic. Essentially, it makes it brittle. Again, not to be concerned with unless it starts to fall apart, crack, warp, etc. Just be certain the filter cover properly seals against the base so the air is "cleaned" before it enters the engine.
Regarding the oil in the oil filter and base, it is supposed to be there. I am a Honda dealer and trust me, these foam air filters (part # 17211-899-000) are shipped with excessive oil. It is not unusual for the oil to "drain" to the lower part of the filter. They are packaged in a sealed plastic and there is a visible quantity of excess oil in the package.
But you may ask why the oil in the filter? The oil in the foam filter traps the tiny dirt particles before they enter the engine. In a dusty and polluted setting (even pollen) the oil soaked filter will greatly extend your engine's life.
It is perfectly ok to wash the filter out with detergent and water (this is actually recommended by most manufacturers) but you need to saturate the filter in oil again before reinstalling it and running the engine. For about $8.00+/- you can replace your filter if it starts to deteriorate. If it gets worn, brittle, excessively stained, or otherwise damaged, REPLACE IT! A brittle filter will fall apart and suck into your engine, causing rapid internal engine damage. To test for it being brittle, squeeze and release a portion of it between your thumb and a finger. If it returns to its normal shape than it isn't brittle. Just remember, for about $8 you can replace it; is it worth taking chance damaging your engine with a questionable air filter? (Your engine should consume about 3 gallons of air per SECOND, a little more than 11,000 gallons per hour, at full RPM.)
Find your Outdoor Power Equipment owners manual here
Here are the directions, from the Honda Owner's manual:
Air Cleaner Service
A dirty air cleaner will restrict air flow to the carburetor. To prevent
carburetor malfunction, service the air cleaner regularly. Service more
frequently when operating the generator in extremely dusty areas.
Never run the generator without the air filter. Rapid engine wear will
Unsnap the air cleaner cover clips, remove the air cleaner cover, and
remove the element.
Wash the air cleaner element in a solution of household detergent
and warm water, then rinse thoroughly, or wash in nonflammable or
high flashpoint solvent. Allow the air cleaner element to dry
Soak the air cleaner element in clean engine oil and squeeze out the
excess oil. The engine will smoke during initial startup if too much
oil is left in the air cleaner element.
Reinstall the air cleaner element and the cover.