1) If you have an E53 (I'd recommend you add the year, model, etc.) to your signature - very helpful so we know for sure some specifics.

2) IF your E53 has the digital / automatic climate control display and you have NO idea if the FSR / FSU was ever replaced, replace it. Your symptoms would indicate that it is time. You do NOT want it to fail when it's cold in December (my experience). The part has been updated - the current design has a plastic base with 29 metal prongs as a heat sink. Original all metal with about 11 prongs. The OEM to BMW is Valeo (I have this in my records). You'll find funny parts offered - I'd endorse a genuine BMW one or Valeo. Hella is simply rebranded Valeo.

3) Your other issue indeed sounds like low voltage, you need to test alternator output (well above 13 V even at idle).

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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.