There will be some side-to-side, up-down play at the slip yoke into the TC. The yoke is just a slip fit (hence the name) over the splined output shaft in the TC. There is a bushing (a thin metal sleeve, really) that rides over the outside of the yoke. That bushing can wear out but 1/16" doesn't sound abnormal.
If you are getting grinding noises, I would pull off the DS and take a look at the output shaft and the yoke. Make sure there is no evidence of abrasion or rubbing. Before you reassemble pack the inside of the yoke with grease.
There should not be any up-down or side to side play at the pinion, however. If you can feel play here that inidicates a worn pinion bearing. Bad bearings here are usually accompanied by leaking from around the pinion seal, so if you're worried about this clean off the front of the diff housing and monitor it for leaks.
Rotational play in the drivetrain is called backlash and is normal, to a point. It comes from the spaces between teeth on the differential gears. The amount is small on new gears and increases with wear, but usually isn't a problem.
What you don't want is rotational play between driveshaft compenents. There should not be any play between the DS and pinion, or between the DS and slip yoke. That would mean bad u-joints.
Besides all of that, is your DS geometry OK? Are you lifted at all, have extended shackles, or anything of that nature? Is the DS balanced (maybe a weight came off)?
Have you checked fluids in the TC and differential?