Well, I jacked up the back of the car and grabbed each tire and gave it a good shake. Zero end play. If I grab the top and bottom of the tire, it moves just a little. No play if I grab front and back of the tire. If I rotate the tire forwards and backwards, it does seem like a fair bit of slack in the diff, but I'm not sure what's normal. What's that called? Backlash? How do I check it?

there will always be some slop there, does the grumbling vary when turning? like louder/quieter when turning left vs right vs going straight etc?

Not such that I can tell. It's kind of a whiny growl, if you know what I mean. Not really a grumbling at all.