The bit about the mileage showing even with the key out, would tend to suggest something else is amiss, rather than the door itself. I can't identify the part you've shown - and it doesn't look like it'd be more complex than a shim or a washer for something (or perhaps even something dropped down the door at some point in its life).
I suppose it's possible the remote keyfob has just become 'de-synchronised' from the car. Long shot, but have you tried pressing either of the two buttons on the key FOUR times in quick succession (whilst holding the key in close proximity to the car, obviously)? That should re-sync them, if it has...
I'm wondering whether the 'beep' occurred because you just happened to catch the locking-plate cable while the door was open, which would've triggered an attempt to lock the car - and of course, the door being open would cause a 'mislock' and the beep. From there - dunno - maybe somehow you shorted something, perhaps, and/or caused the key to lose sync? Certainly I can't explain why the loss of that tiny circle of plastic would've caused this - doesn't make sense...
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