Momus . The key & lock issues you have - I had exactly the same issues and symptoms. I previously used the key to unlock the door tumbler, but the lock cylinder itself broke after 1-2 years of daily use (rebuilt with a kit). This prompted me to check the two issues: keys or the FZV module.

I sent one key to the GM5 guy who redoes them (battery was dead but it didn't fix my issue). www.bmwgm5.com

In my case it was the FZV module (inside the rear hatch, you remove the upper plastic cover) that was defective. I bought a used FZV (check the frequency as there are different once depending on where you live in the world). Easy to replace (no coding required) and it fixed my problems.

As for the other issues you describe, some are simply due to the age of the car (headlight lens hazing) vs. others due to endemic E53 replacement items.

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2002 Euro X5 (E53) 3.0 diesel (M57)
5 speed manual with UUC short shifter.
Pre-xDrive NV125 transfer case (chain & gears).
Factory Xenon headlights, Spyder LED tail lights
Updated vortex crankcase breather.
Motorsport dead pedal, factory fire extinguisher, factory first aid kit, factory F&R mud flaps, factory PDC

Bilstein B6 shocks F&R
Redline oil has replaced "lifetime" fluids (F&R diff, manual transmission, transfer case and P/S).
Dimple magnetic drain plugs in all compartments.