Oil Leak/Burning Smell in Car?

I have a 2012 Honda Civic Si. Early yesterday morning I noticed a burning smell coming from the HVAC when I had the heat on, later that night it became stronger so I checked my oil level; oil needs to be changed and is LOW. I can't get underneath it at the moment so I poured some more oil in for the time being. I checked the undercarriage and noticed that near the filter and drain plug is covered in oil. Any idea what this could be? I d rather address the problem before I have to toss money to the mechanic on Monday.

TLDR

Burning smell in HVAC, overdue oil/filter change, oil leak covering undercarriage, drain plug, filter.

What is the culprit and how can it be fixed?

Added (1). Update: Burning smell comes from inside and outside car, maybe it's oil burning off the engine?

Maybe a loose oil filter or bad oil plug gasket seal. Oil is dripping onto the exhaust system and burning off. Oil plug gasket is a soft metal "washer" that manufacturers suggest one replace each time the oil is changed, but rarely does anyone do it. Most likely whoever changed the oil last did not tighten the filter enough. There's a fine line between too tight and too loose. You will have to check both.

I suggest that you have the oil and filter changed and while it is being changed, the problem "MAY" go away if it has anything to do with the oil plug or filter. Make sure that a new oil plug "washer" gasket is used on the oil plug.

I once owned a 1949 Ford pick up truck. It had blow-by so bad that the oil was being blown out of the dipstick hole. Unfortunately the dipstick hole was right over and exhaust manifold. I left a blue-black trail of smoke behind me when I ever drove it. I had to keep at least 2 gallons of oil in the back of the truck in order to make it anywhere and back home. I didn't have money to fix the engine, so I put a bit of fuel line around the dipstick hole and re-routed the blown out oil down below the exhaust manifold so that I wouldn't make a smoke screen behind me and attract "fix-it" tickets!

Could it be the drain plug or filter? Possibly but leaks can start on the top of the engine and work there way down. If the oil gets on the exhaust it will make the burning smell that your smelling now. The prime way to find the leak would be to clean the area and use a UV leak detection dye kit. Go to a dealer or independent mechanic but NOT a place like Jiffy Lube, Pepboys or any big service chain which are notorious for bad service and screwing peoples cars.

No way to know without looking at it. Better keep an eye on the level, if you let it run out your engine goes bye-bye.
Oil is getting on an exhaust pipe and burning.

Just let the mechanic let you know where the oil leak is coming from. I hope you don't think an oil lube boy is qualified to do more than change the oil! I'm talking about a real mechanic.
It could be a leaking valve cover gasket or an oil pressure warning light sending unit that's leaking.

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