I have a Ford 3.8 that has a leak, it looks like it was

Check your head gasket.

There are some simple kits which have a glowing fluid which can be seen with a UV ultraviolet light supplied in the kit.

You add the glowing liquid to the engine, run the motor, and inspect the drips for the leak location. Then you fix that bad part.

Now sometimes the sending units do leak. But any gasket or seal can leak just as well.

Do not run the engine with the oil light on. Check your oil level constantly. No oil=no engine.

Can not speculate on battery problem from info provided.

It is very unusual for oil to run from a water pump, but you can get a leak from the grease that lubricates the bearing. If you do have what looks like oil or grease, you probably need a new pump.

OIL LEAKS UNDER CAR COULD BE COMING FROM LEAKING OIL FILTER OR OIL PAN GASKET.CHECK FOR OIL LEAKS AROUND REAR MAIN OIL SEAL AND FRONT TIMING COVER CRANKSHAFT SEAL LEAKING OR LIP SEAL AT THE LOWER TIMING COVER TO OIL PAN.

Maybe, maybe not, It's most likely a bad rear seal or worn main bearings but check the oil filter just to be safe. Sometimes things go wrong when changing the oil filter that can cause similar symptoms.
1) old gasket left behind.
2) cross threaded filter.
3) loose filter.
4) damage gasket or defective new filter.

All you have to do is get under the car with a 1/2" drive ratchet, 3" extension, an extension swivel, and the socket for the bolts on the oil pan. Go through all the bolts and tighten them all up. You will be surprised at how loose some of them are.

there are frost plugs on the engine block behind the bell housing, bu first make sure that the plugs on the heads are not leaking, removal of the trans, is no fun but may be the only solution

Check around your rocker cobver and around the engine for oil leaking down over. it is possible that its just oil sat in the catch tray under the car leaking now and then. I would expect after 2-3 weeks though that all this would be gone.

I always pressure wash the oil leaked area and then when it starts again you have a better idea where it came from.

regards
dan

clean engine with degreaser then run it to locate the exact location for leak

yes the 3.8 uses an external oil pump.