It is possible to run an entire script on the Linux side? If so, how, what and where to do that? I am new to using Arduino Yun and I would like to know if it is possible to run an entire script on the Linux side. In my case the script should execute a HTTP request to a remote server, parse the subsequent HTTP response and return some value that then can be used in the Arduino sketch.

If it is possible, what I have to do? For example, how should I write and run script files? what programming language should I use? where I should put those script files on the Linux side?
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.