That diff is covered in a fine mist of diff fluid. It's not leaking out fast enough to make drips, but enough to cover the exterior of the diff, then mix with dust etc. to make a sticky coating over the whole diff body. Not sure which of the three seals (front, left and right) and causing it but at least one of them is gone. Clean down the diff, change the fluid and watch for the leak - one location will get "dirty" quicker than the rest.
Re. the transmission, the consensus is the fluid flush (100%) will likely kill the transmission. A drain and refill (with a filter change) will change about 40% of the total fluid volume. Do a second drain and refill while later and you should have mostly new fluid in service. This method reduces the chances of crap getting into the valve body, and you can change the electrical connector seal when you do the filter change.
But it may be that only a mechatronic rebuild will fix the hard shift.
