It is possible to run an entire script on the Linux side? If so, how, what and where to do that?
Arduino Yun ships with python pre installed. You can copy python scripts from your pc to the yun with scp (or WinSCP or putty, if you're on windows). Once copied, just run it using Process from your sketch. As they are scripts, program name is python and the path to your script is its first argument (you can do that in a different way, but that would require some linux magic, which I suggest to postpone if you're new to it).

Other pre installed options include lua and bash. Other languages such as ruby, erlang and php are available as optional packages.