Most rear ends get a little slack in them after break-in and will act like this. Could be bad u-joint, or mis-aligned carrier (causing slack between ring and pinion.) Probably not a limited slip problem - when the axle clutches wear out they just quit spinning the tires.
Bad bearings usually make a grinding noise. Axle bearings will be louder when turning a corner or swapping lanes (when more weight is on them.) Carrier and pinion bearings have a more constant noise.
I'm in the process of rebuilding the 8.8 in my Cobra right now. Rear had been sounding like a washing machine. Got in and found the driver's side carrier bearing was shot and there was metal all over the place. Excellent time to upgrade to 4.10s!
As for the leak - you can't replace the pinion seal without pulling the pinion, replacing the crush bearing and re-torquing everything. Might as well replace all the bearings while you have it apart. Not sure if there are any reliable additives that will stop the leak.
Highly modified 67 coupe, rusty 68 vert, 03 Sonic Blue Cobra Convertible, 03 King Ranch daily driver
(not pictured) stripped 65 fastback waiting to be a street machine