I have a 2006 RL. It burns 3 to 4 quarts of oil per month. Blue smoke out the exhaust when I start it. EGR valve gets carbon caked in it and I have to clean it out every couple months. Now is also throwing P0300 through P0306 codes saying each cylinder is misfiring. Engine runs rough. Is this related to the oil burning? Engine codes started suddenly and the car was running fine previously.

 

ouch, if its burning that much oil your plugs are probably pretty well fouled

burning all that oil isn't good for the cats either, they'll probably die on you as well before too long

 

According to your symptoms you have oil leaking into the cylinders. Might be your piston rings are fried but definitely a seal/gasket/ring is the problem.

HAve you tried running a compression test on the cylinders? That would help you isolate the problem to a specific location.