NCB: I would assume that, as long as you walk in during the Muttar time, the door sensor is considered a Psik Reisha D’lo Nicha Leih. That’s the only thing I can think of.
About letting a non-Jew push the buttons. I think the theory is that when you get in, you are doing nothing to the elevator vis-a-vis your weight. Sure, a Melacha will happen one the elevator starts to move, but that’s a Grama at worst because right now all you’re doing is standing there. If you walked in after the Goy pushed the button and while the elevator was about to start to move, it would be more of a problem. But if you’re in there before a button is even pressed, then it’s for sure nothing more than a Grama by being Shev V’al Ta’aseh when he pushes the button.
I have been told that R’ Schachter and many others are Meikel using a Goy to push the elevator if there is a significant number of stairs. The logic is that it’s a Shvus (Amira L’akum) Dishvus (electricity) BeMakom Oneg Shabbos because walking up a ton of stairs is a lack of Oneg Shabbos. R’ Schachter has also said that it’s Assur to move into a building Lechatchilah where you know that this will be the case.