1. Yup model A1339
2. hmmm maybe a clue. it shows up in the blue tooth indicator but does not show up in the system settings, only the mouse, keyboard, and printer(s)
3. no corrosion visable
4. the only switch is on the starboard end if the battery compartment, and it flashes the green light indicating it is looking, but never connects.
5. no cables
6. na
I will update the list, however everything but the system software is pretty much the same, currently running os 15.4.1 on both the 27 inch iMac and the mackbook air.
I am now thinking that since it all worked before we left for a few days and I do not recall what os was running then, that an upgrade happened and now 15.4.1 might have a bug, OR the dead battery(s) caused damage.
I do recall when you made the switch to the magic trackpad way back when. I had been using it for quite a spell then.
daryl