(Klunking diagnostics) Coils? Struts? Can push the wheel up when it's on a lift?

556 834 is the correct Sachs part number for the Advantage e39 sport replacement strut.
557 210 was probably the old Sachs part number for the OE BMW sport strut.
31 311 096 857 is the BMW part number. It has no bearing on the Sachs part #.

Good to know I installed the right part. Thanks!

Originally Posted by EconoBox

FYI, they said they never replace the strut mounts, and they just repack some grease into them.
(Mount bearings don't really spin at rpm like a belt pulley or wheel hub bearing)
But, one of my mounts' bearing was a little "rough" b/c the protective cap was missing.
So, I replaced both strut mounts, since I had ordered them anyway.
.

This is poor advice from your service tech. The rubber insulation w/in the strut mounts are subjected to repeated force and they develop hairline cracks that are invisible to see. however, when you go to push the bearing away from the mount, such as simulating the damper ramming upwards, that's when the cracks can be seen. IE, you have to pull the bearing away from the mount in order to see any damage on the rubber insulator. But it's good that you chose to replace them nevertheless. The other question is this: did you replace your spring pads too?

Yes, I replaced the spring pads even though they looked perfectly fine.

One thing I've learned in the last year is that I think there is a LOT of "forum paranoia" that real mechanics just laugh at in the real world.
We definitely waste tons of money replacing stuff that doesn't need to be replaced, just because one guy on the internet said you have to.