Do you know if the shop did a leakdown test on the engine cylinders, and if so, what the results were?
Did the shop retreive the fault codes for the check engine light being on, and if so, what the fault codes were?
It does sound like a loss of compression, which combined with burning oil, is either worn piston rings, or worn valve seals and guides. The leakdown test on the cylinders will let you know which it is. Another way you can determine is to do a compression test and write down the compression of each cylinder, then pour an ounce of engine oil into each cylinder and then do a compression test again. If compression comes up, then the pistons are worn. At this point, you would need to either do a lower engine rebuild or replace the engine. If compression remains the same, then the cylinder heads need to be pulled and a valve job done. This would run about $1500 for the valve job, parts and labor.
Most of the time it is due to worn valve seals and guides, so I would have a shop do either of the tests above to verify source of loss of compression and burning oil.
And yes, you can go to Autozone to have them scan the computer and tell you the engine fault code numbers and definitions
Yes, with the fault codes, I could tell you what you are looking at to diagnose and repair that, but most likely this will be a different issue from the low compression and burning oil.
For a delaership to do any work, you will be paying more for parts and labor. Labor to do a valve job is 14 hrs, and parts will run about $150.
At average dealer rate of $125 per hour, you would be looking at $1750 in labor, plus the $150 in parts for total of $1900
At average independent shop rate of $80 per hour, same job would run $1120 labor, $150 parts for total of $1270
For engine replacement, at dealer 27 hours- $3375 for short block labor and $3300 for short block, total $6675
Independent shop can drop a complete remanufactured engine for 15hrs labor - $1200, and $3000 for engine - total $4200
Ok, the random misfire would be related to the low compression as low compression will create engine misfires. This is a good indicator, although not conclusive in and of itself, that it is a valve problem. The lean code is normally due to a vacuum hose leak, and sometimes an intake manifold gasket leak.
If the engine has a deep knock, this could be an indication of lower end engine problem. If it is more a metallic or high pitched knock, then likely the valvetrain.
I would at least have the leakdown test performed. If the compression leaks down on this test, then you are looking at the valve job, and I would go ahead and have it done. If however it indicates worn piston rings, then I would lean towards trading it in.