What you can try is warm up the engine slightly and spray it down top to bottom with a can or two of engine degreaser. Let it sit then spray it off with a garden hose or light pressure washer. Just try to avoid major electronics like the distributor, alternator, and computer, just get the engine. After rinsing let it dry and start it to dry out the rest. If you plug wires are not really old there shouldn't be any issue if they get a little wet.
That should be enough to see where any fresh oil is leaking from.
On the 5-speed that thin plate is just a cover plate to keep dirt, mice and cats out from around the flywheel. The automatic looks the same in the pic but someone else needs to verify that. Nothing structural and it lets oil or water drain past it. That spot in the photo is where oil will drip from and it could be from the front, top, sides or back of the engine and getting blown back there. A few drops can travel a long way.
If the engine is clean and it's dripping from behind that plate it could be the engine rear main seal or the transmission front seal. I don't know about automatics of if torque converters leak. That plate does come off by removing the bottom bolts and it slides out.
Engine oil is usually black unless it's clean new oil and transmission oil looks new but has different colors, gold if oil or red if auto-trans fluid. You'll need a white paper towel to help decide the color.