Okay here goes:

Do I have to remove the axles from the carriers to get the differential out of the car?

How can I separate the diff from the axles and do the least amount of damage?

 

Yes you have to take out the axles from the diff and the hub assemblies to take the diff out of the car. I honestly think that was one of the biggest PITAs in my auto-manual swap. All I can remember is that it involved a rubber mallet and some banging.