Oil getting into the air filter and cover after a ride

Air cooled engines run hotter than automotive engines. The use of non-synthetic multi-viscosity oils (5W-30, 10W-30, etc.) in temperatures above 40 ° F (4° C) will result in higher than normal oil consumption. When using a multi-viscosity oil, check oil level more frequently. for your mower you can use W30 if you are above the 40 F
clean or replace the air filter
replace the fuel filter
grease the axle
sharpen the blades
if you need a manual you can go here.
Best of luck

A dirty air filter will reduce the power produced by the engine, and eventually may make an engine stall, but it does not cause it to "burn oil". You have options, so just work your way through them: 1) manager/owner of the repair shop - will they stand behind their repair work? 2) the warranty and protection plan from Lowes. Go in with the right attitude - don't direct anger at the person you're talking to, just describe your issue and ask them what they would do in your situation? If you get turned down, politely ask to escalate the issue to their supervisor, etc. Good luck to you.

remove the side cover
undo the nut in the middle and remove filter
take the foam of the frame
clean with a solvent, kero or similar
clean again in detergent and water
continue until spotless
let dry
use a air cleaner oil from a bike shop
put some oil on the foam and squeeze until oil is completly over filter( will be all the same color)
can do this inside a plastic bag to save the mess
fit back on frame
place a smear of grease around the edge that has to seal.
refit in air box, with centre bolt
feel all the way around to make sure it is sealing.
refit side cover.

doesnt matter too much if it is still a little wet
do this after every dusty ride
even twice on very dusty rides

I keep about 3-4 spare filters, clean in plastic bags, and wash them all together.

dont forget oil changes as well, they hold a bit over 1 litre, so change it often

This is probably nothing more than oil mist which has gathered, I believe it is part of the system to scavenge oil mist from the crankcase and it is routed to the air cleaner housing to be burned. Sometime at higher RPM's the system, marginal at best, is overcome and what you found is the result. There is really nothing to worry about, the system is working as it should. FWIW Harley Davidsons are notorious for doing this, and it is not uncommon to have the whole right hand side of the bike covered in oil after a fast ride.

I could be caused by a couple things, the most common is overfilling with oil, running on too great of angle and flooding the valve stem area or really bad valve stem seals.

All Harleys after the year 1992 will have some engine oil carry over visible in the air cleaner though there is a diffrence between excessive and normal although if you get on your bike and travel down the highway at a constant 80+ MPH you will find that you get more carry over to the air cleaner then you would normally riding at lower speeds and this is simply cause you are running the engine harder and therefore you will see more carry over at the air cleaner. Now to be sure, make sure your bike is sitting on level pavement and in the case of a Softtail model it should also be upright and level pull the dipstick from the oil tank and wipe it clean, reinsert fully into the oil tank and remove it again and look to see if your oil level is roughly halfway between the check marks on the dipstick. If so then yes your oil level is good. I would now remove the air cleaner cover and wash any excess oil out of it and if your running a stock air filter and you have had a oil carry over issue for awhile now, then replace the air filter, reinstall the air cleaner cover and go ride the bike for around 50 miles and by ride the bike I mean go ride it like in town and on the highway without being abusive to it or excessive speeds(70 mph). Then remove your air cleaner cover again after your test ride and look to see if you have any oil carry over in the air cleaner. if so let me know and we can go into further detail.

This is normal on the sporty. I rode a 1200c for many years and from time to time some blowback would accumulate in the filter and drip from the bottom of the filter cover. Just clean it off and enjoy the ride.

You have a leaking fuel valve in the carburetor bowl.

oil seems to be comming up from the crankase breather tube, which shoud only bring up fumes. most probable cause is oil is over filled.if oil level is good, problem lays in the breather valve mechanism in the engine, or severly leaking piston rings

Sorry to say this is a enteral problem with the engine. Either valve guides or piston rings. Best solution is to buy a new mower, unless it is a expensive riding mower then you might be able to repair the engine for less than the cost of a new mower.I know you don't want to here this but I do believe it is the only way to go.
Kurt