I thought the single click like you describe means the solenoid IS working, but the contacts to power the starter itself are worn. I've seen it on my old Trooper and on my father's old Toyota PU (which both had the exact same starter motor).
That starter motor had a copper contact on each side of a copper ring that was attached to the solenoid shaft. When the solenoid activates and pushes the gear into the flywheel it pulls that ring into contact with the two copper contacts and juice flows to the starter motor. The copper contacts were worn down in both starters, and replacing them fixed it. I got replacements at a local alternator/starter rebuild shop. In both cases the symptoms were identical to what you're describing... the starter symptoms that is... I don't know about the battery and transmission problems.