I am not up to date on your other thread, so maybe you can bring us up to speed.

There seemed to be confusion on whether they installed the third member.

Or just the differential.

If they installed the 3rd member that means you should have your other one in your possession, complete and ready to go. If you have a "decent" collection of hand tool with metric sizes, and "some" mechanical ability you should be able to put your original third member by yourself.

This would confirm if it is the third member or not.

They are correct just pulling the drive shaft will not cause the tranny to lose any of fluid. Unless it was overfilled it usually wont lose any gear oileither . However if the truck is driven with the drive shaft out, the yoke is not there to hold the gear oil in, and you will lose a good deal of it.

IF, that is an if, if the fluid was low, and they aren't bullshitting you, it would be hard to prove that they did it. This is a manual or an automatic. I am guessing a manual.

There is no extra stress from the LSD. The LSD in a straight line will behave the same way any other diff would, open, locked, welded. In a straight line they all do the same exact thing.

Make sure you document everything that goes on on paper. You might have to take them to small claims court, and you dont want it to be a she said he said thing.