Glad you found it fast.
As for the trim...
Have you measured the output to the trim motor without the trim motor attached?
Was there clicking at that time or only when the motor was connected?
If you are measuring the voltage going to the trim unit at the connector, does the voltage drop when you try to activate the trim.
The clicking could be a bad trim relay unit, but it could also be a bad connection or bad connector where it connects to the battery post on the starter or a bad connection at the connector near the trim relay unit.
You should also examine those ground wires for points where they may have melted or where it connects to the block. Make sure the connection is tight and good and clean.
The clicking sounds like a voltage drop when the trim motor is trying to draw a high amount of current. The no load voltage may read good, but under load, if there is a bad connector, wire, etc, the extra resistance will cause a voltage drop. The clicking may be happening by the first click of the relay being activated, but when the load of the motor is present for that first instance, it causes a voltage drop, which causes the relay to drop, releasing the load. Since you still have the switch pushed, that reactivates the relay again, adding the load, causing the voltage drop, dropping the relay, etc etc, etc, over and over again....thus the clicking.
Let's see what the results of you load and no load tests are...and we will go from there.
Again, if the motor was off and wires disconnected, they may not have been reconnected correctly. In general, the way you described the original short of the battery cable, it should not have hurt the trim unit....except for maybe melting the ground wires.
Mike