To answer your questions (LMK if I missed any):
1. If I remember correctly, and if I don't someone will correct me, you can engage the center diff lock at any speed below 35mph. Unlike shifting the transfer case gears, you're not moving gears from engagement to disengagement, you're just grabbing both driveshafts and holding them so that they turn at the same time.
2. You have an automatic transmission that shifts into 1st gear when you slow down and stop. So, when you're stopped, and you engage the axle locking motors, the transmission will be in 1st gear. I can't remember if the shift lock computer will allow you to engage the motor while the truck is moving, but you shouldn't do it. The gears inside the differential could be moving and if they hit the lock pin, that would be bad.
3. The axle lock doesn't care what the transmission is doing; it's just sticking a pin in the spider gears to keep them from rotating around the side gears. It does matter what the wheels are doing, because when they are spinning, so are the side gears. (I have a question ) I'm just a little confused about what is happening when I put it in low vs high that it stays in all the time . If I'm understanding you correctly when I'm in High which in my truck is always in 4 wheel drive , all 4 wheels are turning all the time but if one wheel ,say the back left side gets lifted off the ground that wheel is going to turn and cause the opposite wheel to do nothing but the other 2 front wheels will be turning so its really a 2 wheel drive , but when I put it in Low what is changing as far as traction goes if I don't engage the lockers ? That's the part I'm not understanding . When I put it in low without using the lockers is this action somehow causing the back right rear to turn even if the left rear is still of the ground and spinning ? Or does going into low just give you more torque and it allow you to switch on the lockers either just the rear or front and rear together ? One more thing, If I do add the high center diff button ,once its installed and pushed ,will this allow me to use the front and rear diff locking knob ?
this is what the inside of the differential looks like:
View attachment 3343201
The transmission turns the transfer case gears, which turn the propeller shaft (AKA the driveshaft - this is what they greased for you), which turns the drive pinion, which turns the ring gear.
The ring gear is bolted to the differential case, which has the side and spider gears in it.
View attachment 3343209
All of this fits inside the differential carrier you see in the center (almost) of the axle when you look under the truck.
You do not "need" to install a manual override switch. The system works just fine as it came from the factory. As a matter of fact, if you're not going to drive offroad, on trails that are difficult, I wouldn't bother at all. It's up to you whether you want to take the time to modify the center diff lock control so that it's lockable in HI. My advice would be try to drive the truck as it came from the factory, and if you find it needs more, add more.
I am having a hard time believing too much grease in a slip joint would be the root cause of the problem you described, but I guess anything is possible.
I'm sure you can find someone in Ashevillle who's worked on a Land Cruiser once or twice. I'd be surprised if you can't find someone where you are that can do it. If all else fails, take it to the Toyota or Lexus dealership and ask them to look at it. It won't cost you a thing.