My understanding as a Jew (though not Orthodox) is that you are forbidden from manually turning electrical devices on and off, and from altering the behaviour of a running device. So if you, say, accidentally turn on a stove just before the Sabbath starts, you then cannot turn it off until the Sabbath ends (or you get a non-Jew to do it for you).
> An elevator that cycles between floors and one that only moves to a floor to pick up a passenger when they wait in front of the door might be indistinguishable from the perspective of the user. Would the difference matter?
IANA rabbi, but I suspect this does indeed matter. In the former case the elevator is just doing the same thing over and over again, while in the latter case the user is actively controlling the elevator.
> What about showing the intent to exit the elevator at a particular floor by e.g. taking a step towards the door of the cabin and having the elevator detect that?
Again, I suspect this would be forbidden.