Rear motion from left to right

Check the suspension on that side. It could be a damaged spring or a blown shock absorber. Get the suspension linkages checked as well. There might be a loose or worn bush or mount.

rear brakes are sticking get it serviced

Could be a multitude of problems.
suspension drop links
anti roll bar bushes
ball joint
suspension mount
gearbox mounting
exhaust loose
wheel bearing
suspension strut
drive shaft

Best to get a mechanic check it to determine the problem as it could be any of the above.

A sway bar is a suspension piece that keeps the left and right side of vehicle connected to help with turns. They are held together through sway bar links at each end, which are fairly common to break or rattle over bumps.

check if rear differential is straight , or remove 2 rear susp coils and compare length., also check the rear arms
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Blown suspension. You need to replace the spring and check the suspension spring supports. Also check all the others front and rear.

the speedo is electronic, not cable, this problem is caused by a sticking stepper motor that moves the needle, a stepper motor is a miniature electric motor that can move and hold one position, this was a common problem at the dealer I worked at, u will need to pull the speedo and send it out for repair, I use the following link for this service.
http://www.dnaspeedometers.com/

check to see if there is a ride height sensor in the rear of the truck and make sure that both side are even.

you didnt say if it was 4x4 but here is a pic
can you7 tell me what socket is out of joint?

If it's wheel bearings, they should get louder the faster you go. Also, if you have one going out, loading up the suspension on that side will make it louder. For instance, if you have a bad front left bearing, and you take a right-hand turn at speed on the highway, that will load the left side suspension since the car will roll to the left. The noise would get louder. Similarly, if you took a left turn, and the weight shifted off of the left side, the noise would go away.

Try jacking up the car and grabbing the wheel at the top and bottom and shaking it hard. If you get any play or looseness in the wheel, chances are the bearing is bad.