Not sure where you got the magic number theory, is that like the single bullet theory?

Depending on incline, the trans may take a minute to engage going in reverse uphill. Common practice is you keep a foot on the brake to you feel it grab.

As for the doors, could be a bad battery in your key not sending a signal every time and for latching, have to explain "close loosely" a little better.

Losing acceleration is entirely subjective. If you think there may be a tranny issue, take it to a dealer and have them run a diagnostic on it. While water pumps frequently go in ALL BMW's, transmissions do not.

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