The symptoms you describe match damage to the module (gm3?) That controls those items.

The short that shows the tailgate button pressed is upstream of the button (module or wiring loom).

Determine which module controls the items acting up and you'll likely find the source of your headaches. (and thank foxwell for finding the bad part)

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)