There is play in the front drive shaft

The green ring is just a nice colored seal protector. It covers the transfer case seal to keep crud out of it, The rotational play in the shaft is common gear wear and some is needed to allow the 4X4 to operate easily without binding, although you can have an excess amount. His is getting close to the point of needing some work. The play shown in the CV joint where he comments about the noise he hears when driving however is not normal, that is a constant velocity joint and Jeep has had issues with them on the drive shafts for a long time. Used to be that you only could get them with a new shaft but now they make rebuild kits for them that are easy to use and come with instructions.
Rebuilding them is actually quite easy as you just remove the bolts around the joint and take the pieces all off. Then clean up the splines on the shaft and any rust from the yokes. Then install the new parts and fill the joint with the included grease, then bolt them back together. Both ends are serviced the same way. If you have the in out play at the front joint like his where it moves in out with a clunk it is getting worn enough to be repaired.
In out play on the other end at the front axle is normal as it is designed to slip as the suspension moves, that is why is has the long boot.

That's normal wear on yours, all internal to the differential and transfer case. The gunk you see is the grease from the universal getting out, It would be okay to change it out but unless it's loose, binding or noisy it's still okay.
As far as the backlash, you would need to rebuild the differentials with new gears and bearings as well as new spider gears to remove all the wear, and even then you will only remove less than half of what you have now. Personally I wouldn't do it until it's needed. Or if you were planning on doing something like an engine swap and lift kit and other suspension upgrades, then you would be changing the gear ratios and doing much of that work for the project anyway.