That ring gear is beautiful. Good pattern. If you can't measure the backlash (btw, you need a dial indicator, not a micrometer for that), try turning the ring gear with a breaker bar on the bolts that hold it to the carrier. You should feel a very slight click as you move it back and forth, but it should move too little for you to be able to see it, relative to the pinion. You should be able to feel it contact each side of the pinion gear, but should not feel sloppy. My guess from looking at the picture is that it's already set up much better than most are, unless you've polished up those teeth somehow.
I'd start it up with the cover off and no oil in it. put the trans in low with both rear wheels off the ground and the wheels & drums removed. Don't make it go too fast, or do this for too long, 'cause lack of lubrication will cause damage over time. You could spray some wd-40 or CRC lubricant on the ring gear, that'll help.
Take a solid metal rod about, say, 18 inches long and use it as a stethiscope. when the axles are turning and the ring gear is rotating, stick the metal rod up to the inner axle bearing retainer caps, one after the other, with the other end stuck up to the bony place right behind your ear. That'll let you hear what's going on fairly precisely. do the same thing with the outer axle bearings and with the pinion bearing at front part of the diff, behind the u-joint, on the third member (sticky-out thing that houses the pinion shaft). See if all those bearings sound ok. YOu shouldn't hear any grinding or rumbling, just a smooth whirr from each of those spots. Any of those bearings will make the kind of whining noise you mentioned if they're going bad.
If you can dismiss each of those as the culprit, I'd guess that the whine is coming from the transmission. Did you see that it has fluid? The power steering pump can make the same kind of sound if it's low, too.
Btw, you do have 4wd disengaged, and the hubs unlocked, right?
