I've never tried removing a diff from the top, I'd say its impossible.
Seeing how the truck is in the mud, my first suggestion is to have the autowrecker pull it out for ya, most will do that for a minimal charge. The little you will pay is worth it imho. 8)
If they won't do it, here is the steps in a nutshell.
First off
It will be much easier to remove the diff without the engine in the vehicle.
It does make it easier to remove it if the vehicle is supported by the control arms, this spreads them out a little and gives more room to drop it out.
Since you won't be using the passenger side tube and shaft anyway you can take that off in the vehicle as well, that will make it more managable to work with as well.
It is possible to remove the diff with the steering hooked up but some find it easier to seperate the idler and pitman arm from the centerlink to give a bit more room to breathe.
You will need to swap your right axle tube and input flange onto the new diff for it to work in the syty application.
Hope I answered some of your questions
Tony