BMW E38 steering wheel vibration and shimmy

The most common cause is worn thrust arms (reaction rods), which are often the fault when the vibration comes especially under braking. Other causes include loose lower control arms and control arm bushings, and tie rod ends. At the rear, the large bushings can be loose even though they look intact.

Yes. Bent wheels or a missing centring ring cause clear shimmy even if the tyres have been balanced. A centring ring is needed if the wheel's centre hole is larger than the car's hub.

Warped brake discs or a seized brake caliper can make the steering wheel shake under braking. Cleaning the caliper and lubricating the slide pins can fix the fault. The discs should be measured with a dial gauge before replacing them, because a small warp does not always matter.