You didn't say which battery terminal you touched it to.
If you touched it to the + (hot) terminal, nothing should have happened.
If you touched it to the - terminal it should have stayed on as long as you
made contact. If this is not the case, the relay is not hooked up right
(or may be malfunctioning).

Check the switch with your meter or test light before removing it.
In all gears except reverse there should be no continuity to ground.
In reverse there should be a dead short to ground.
If this is not the case you need to remove the switch and see if it has
something shorting it or if the contact is missing or whatever.

There is no copper ring on the tranny, it is a 3/4" NPT pipe plug, perhaps
with a magnet in it. If it is dripping, you should tighten it up and it
normally will stop. If the engine is dripping it may be that you need a new
copper ring, or you may be able to get it to stop by tightening the drain plug,
but don't strip the threads.

If it is dripping from the bellhousing it is probably your rear main seal.
--
George Downs Bartlesville, Heart of the USA!