It's true that when the input shaft nut works loose the clutch pack will indeed be "wiggly loose". I've had it happen and mine got so loose it was rubbing on the inside of the case cover (to the point it wore completely through). If the nut on the other shaft (with the big gear) is tight then your problem is not likely to be in there. I'm surprised you popped the side cover off already. You had indicated that 2nd gear was working fine so I wouldn't expect to find that nut loose at this time. At least now you know for sure.
An electrical voltage and resistance check at the solenoid valve wiring harness connection would normally be one of the first places to check. I can't remember off the top of my head what exactly the resistance should be across the solenoid coils but they should all be similar values. If you find one that's significantly higher or lower resistance that the others then that would indicate a bad solenoid. The links already given and youtube will have the specifics.
Again, the VSS (Vehicle Speed Sensor) could be the culprit too since the transmission won't shift into reverse if that sensor is reporting a speed greater than 5mph.
I never saw any OBDII codes posted. The OBDII system can actually help narrow down the problem if you can read the codes. I would be very surprised if the SES (Service Engine Soon) light wasn't on. See if you can get the codes and report them here (the code numbers themselves - let us diagnose it here for better clarity).

Update
 

We're trying to help you, but we need all the info available to steer you in the correct journey.

Jamnar is quite experienced with the s series TAAT transmission, and I'm glad he stopped by to pick up the troubleshooting effort.

Do as he requests and you should reach a diag and possibly a fix.

any updates?