You'll need to clean up that area and watch it closely for the source of the leaks. They do have dyes that you can add to the oil, then trace it with a UV wand or similar.
Possible the oil is running down from one side of the engine and dumping off onto the trans. Also possible that the oil leaks only under pressure. Hard to say for sure without being right there.
Common places for the oil to leak is the timing chain tensioner (on the passenger side of the engine, close to firewall). The front and rear oil mains are also common places for a leak.
I'd double check the valvecover gasket and the oil pan. Valvecover gasket is an o-ring type, so it seals best if the material is not completely crushed. There are little metal standoffs that prevent that - but they can corrode over time and it would be pretty easy to overtighten those bolts.
Oil pan uses FIPG sealant instead of a gasket - those generally hold on pretty well. But once they spring a leak, they are nearly impossible to seal - have to pry the pan off and start over from scratch. I's also look at the pan and note any dents or physical damage to the pan. It is actually part of a the structural element of the engine, if it took a good enough hit from debris or going offroad - it would be prime to leak.
But loosing 3 quarts over a 500 mile interval - at that consumption rate, I think you probably have both a leak (from your description) as well as heavy oil consumption. Some of the 8th gen Corollas have been found to consume oil at a pretty alarming rate. Some are 1-2 quarts every 5000 miles, some are 1-2 quarts over 100 miles! There are some that also don't drink an ounce of oil (mine is this way - knock on wood) - but may have other "issues", like EVAP problems (like mine).
Any possible way to get a maintenance history of the car? Any record or documentation of oil consumption?
