First, I would look all around outside the shower and caulk any obvious places where water could be getting by and down to ceiling below.
Then I would look behind the wall where the shower valve is and check to be sure that it is not the shower valve or shower arm leaking (remove faceplate and then run shower without plate on...look inside wall with flashlight and see if any water drips from the valve or from the shower arm above the valve).
Then I might replace the donut shaped rubber seal around the drain pipe in shower if in fact it is a rubber seal (is floor of shower bouncy..?). If disintegrated as you suggest you may be able to pull old rubber out with a pair of needlenose pliers and install replacement rubber (if hard ring around pipe, then drain is leaded in, that will be a little tougher to work on). Is this shower floor a premolded shower floor or a tiled shower floor?
Then I would go downstairs and determine exactly where the drain center is from above (not necessarily where the stain is) and I would cut a 6 x6 hole...(home supply makes a great spring loaded access panel that just snaps into the hole afterward so you can repair hole when you have time to do it)....and then evaluate the situation from there....any real issue should show itself there.
If not, then I would plug off the shower drain and fill the shower pan to the threshold until something revealed itself.
There, that should get you started anyway...;) let us know thoughts...MARK