Not sure about your battery problems, but the alarm sounding at random could be a faulty hood switch, which I have personally experienced in 2 different E53's.

Unplugging the aforementioned hood pinswitches stopped the random alarm tripping. Have not gotten around to replacing the switches, but I read a lot of other people just leave it unplugged.

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04 X5 3.0i auto
03 X5 4.6is