What if you push the button really slowly, so that it's a thermodynamically an adiabatic process? As you release the button equally slowly, it does an equal amount of work on you. The process has been thermodynamically reversed, so all is cool! (The irreversible state change in the button-driven machine, arguably, increased entropy, though, ouch. Maybe not!)
In general no. Non-Jews are not required to keep the Shabbat, but Jews can not ask them to do things for them.
There are two exceptions, one is in the case of urgency that is not life threatening (for example a cold baby in a house with the heat off). (If it was life threatening than you just do it yourself.)
The second is you can hint to the non-Jews. For example say: It's so dark in here. Then the non-Jew is doing it for himself and not for you, and it's permitted. However doing this is discouraged except when it's important.
It depends on your interpretation of rabbinic law. In most cases the answer would be no (since you're still causing the effect by your actions), but some frum families used to employ a shabbos goy [0] to light candles, do shopping, cook, etc. Any action is permissible if it was going to happen anyway, but you get into degrees of technicality of the definition of "what was going to happen anyway".
I have never observed correlation between the intended floor and the actual lift travel and floor you end up on so I'd suggest that all lifts probably operate in a similar manor. Either that or its just malicious embedded systems engineers :)
Go to floor 1
Open doors
Wait 30 seconds
Close doors
Go to floor 2
<Repeat in a loop>