Thanks for the pictures! It looks just like mine , only in a little better exterior condition.
On the trips out with mine I cant hear the differential at all. The tracks make some noise on the sprockets , but minimal. The engine is where the majority of the noise comes from in the cab ,but is not loud enough to bother.
The diferential can be checked for play by taking the u joint off by the diff, and taking the tracks off the sprockets.
check the axcel play and seals behind the sprockets ,turn the drive end to see if there is any play in the bearing end. I was told by a mechanic the play turning back and forth should be a 1/16 turn.As well the mechanic turned mine with a screw driver against his ear and to the diff housing to listen for noises within like several teeth broken etc. The amount these machines are used ,a little play wont hurt as long as the bearings are good.Change the oil.
I am no expert but this is what a mechanic did for me .
48bomer