We've talked/joked about it with the operators before. The site is owned by my organisation and it is my organisations asset (the gate and key fob) so there shouldn't be any issue. I'm not trying to hack some other companies processes, just make our lives easier 
Worst case scenario I'd just have to dismantle it, but it'd be a fun project.
A gate controller like that probably uses "rolling codes". Good luck cracking that.
I'm not sure. There's a whole heap of key fobs, and no obvious RTC battery. When the batteries die they don't need reprogramming or anything. I guess if I at least build a sniffer tool I can see if the data changes over time/button presses?
You may find it easier to put a relay or opto across the local button if it exists,,or dedicate a remote to the task, and ‘press the button’ remotely…across a 433 or similar remote radio link.
I did think it'd be cute to make some over-engineered robotic arm that pressed the button for me haha.