I hate to be the bearer of bad news, but from the looks of the pictures, especially #3 it is the Pinion seal, they are prone to go out when the axle vent plugs since the pressure builds up and that is the weakest point. The Vent leaking will not go forward and cause the build up on the exhaust, that is from a leaky seal and getting flung from the pinion flange itself.
Here is the part # for the Vent Tube fix Version 2 https://www.amazon.com/Ford-DR3Z-402.../dp/B01G2MHO9E
. The seal is the biggest part since even after removing the pressure it will still leak so that and the fluid should be checked by a mechanic.