I leave that one to @Tony Luken.

Good points, I let Tony explain in detail, just a few remarks.
Re 1) "why does advanced camera calibration shifts the center of the displayed image?"
If you backtrace the light-rays that a camera receives, you have
a a fanned-out cone of (reverse) light beams. From all these
"rays" the calibration now picks the one that happens to be
perfectly perpendicular to the machine X/Y plane. Because the rays
are fanned-out at all angles, there is always one with that
property (assuming your camera is not too tilted). That
one ray, or rather the corresponding pixel in the image then forms
the new camera center:

I explained more about that here.
https://groups.google.com/g/openpnp/c/_BqWBSMAdt8/m/rNyOVROUDwAJ
re 2) " If I move the nozzle up and down, its position moves in x and y as well."
The Advanced Camera Calibration calibrates the camera, not
the nozzle. Because we always hold the part into the focal
plane of the camera, i.e. we lift up the nozzle by the part
height, this is also all that we need, i.e. we do not currently
need 3D calibration in the bottom camera. But see (4) about nozzle/nozzle
tip calibration.
If we really wanted to push this further, we would need to
calibration for planarity of the nozzle tip, i.e. how
tilted it holds the part, and how that tilt changes with nozzle
rotation, which might start to matter for very large parts. Not
sure, this would make a big difference in reality, though.
3) "at which Z level will bottom vision take place?"
If you look closely, you see there is an Auto Focus operation in preliminary bottom camera calibration, that makes sure the bottom camera Z is at the in-focus point:
https://youtu.be/md68n_J7uto?t=269
Re 4) " I assume, that best performance (xy-precision wise) is still archived, if bottom vision takes place at PCB-Z."
Yes, it is still recommended. To achieve that, make sure your
camera focuses at that Z height, so (3) will nail it.
But if that is not the case for your machine, its not too bad. With Nozzle tip calibration (a whole different but equally important topic) a bottom camera Z that is different becomes less important, because we have tool specific offsets applied:

5) "Can you disentangle camera tilts from nozzle tilts? "
Tony probably could, but see (2) why this does not make much
sense, and see (4) for how we covered what really matters.
_Mark
> We'd need to make sure that any necessary backlash compensation gets applied before any negative Z motion takes place. Mark - would that be difficult?
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There is nothing as good as having a machine, where everything is
at the same Z (i.e. withing a few Millimeters). Everybody should
really try hard to achieve that.
_Mark
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