vibration at 25-40 mph

If you got rid of the side to side wobble, then the wheel is a bit bent. Look at the new wheel from the side and see if you spot any obvious vertical runout. If not, then it is just the tire shape. It may or may not be any issues; radials and any nylon belted tire take a set when they sit for a while. (Rayon corded tires in the 60's were terrible for this!)

Your 'on and off' vibration sounds like there are 2 sources of vibration; they are phasing in and out and thus reinforcing each other 1 second and then cancelling each other the next. Separate vibrations from engine or exhaust or drivetrain can do this. I can't say that this has to do with the new axle or ring and pinion; seems unlikley unless one of the bearings is worn. Have you grabbed the pinion and tried to move it around up and down and sideways, to see if there is any looseness or play?

BTW, friction modifier makes the sure grip slip more smoothly in normal turns. This is when you will actually have diff chatter. If you are getting some chatter or noise when moving straight-ahead, then that is not what friction modifier will effect.