I've experienced similar issues on my '01 4.4 twice before - both times the culprit ended up being a faulty ignition switch. Dealer replaced ($118 for the part) and all electrical issues were remedied.

Mine never did the "backup beep" you described, but I did have issues with the radio randomly turning off (and always when my gf opened the passenger side visor vanity mirror). Also I had a SES & airbag & seat belt lights on the MID, button lights on steering wheel wouldn't come on, rear wiper would only work in increments (looked like a clock's second hand ticking), and the horn didn't work.

Hope this helps you narrow down your search for the cause!

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2001 X5 4.4i