1. Put some food coloring in the tank. See if you can actually see the spot where water is leaking through a crack in the flush valve/overflow tube. Given your situation, if you can find the spot, I would patch it with an epoxy putty available at any hardware store.
2. Your problem could be caused by a small black refill tube which runs from the filling valve over to the overflow tube. If this tube is inserted down into the overflow, ( it SHOULD end ABOVE the overflow and spray down into it), siphoning will take place giving similar symptoms to what you have.