Parasitic drain from the CD changer is quite possible. Had that in my E38 a month ago. I pulled the cd changer out and no issues now.

Use the cluster and go for a drive, like wpoll said, with lights and things on, and watch the voltage in the cluster. Switch things on and off and see what happens.

I've left my e53 and e38 unlocked on many occasions, but not with the key in them. It has never caused a starting issue, but the key in the ignition might stop the system going fully to sleep after 16 minutes..

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2005 E53 4.4i
2003 E53 3.0d (Mrs)
95 E38 740iL (sold)
93 E32 735i (sold)
91 E32 735iL (sold)
93 E34 535i (sold)
89 E34 525i with M30B35 conversion (sold)