ELI5 Elevators - is the door close button a placebo?

Why does every elevator have a "door close" button if almost none of these buttons actually work? Is it a placebo? Why do some elevators have an up and a down button on their door, when they call the elevator to your floor regardless? If an elevator is falling to the ground, would jumping right before it hits save your life? Why do engineers say a space elevator would have to be built at the equator?

... we explain like I'm five

Thank you to the r/explainlikeimfive community and in particular the following users whose questions and comments formed the basis of this discussion: benignevil, elev8torguy, different_secret4912, ansuz07, captainpatent, flexipiezo, tomrlutong, ravingrook and professorferox

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