Main occupation and paid side activities

Let's say I work as a programmer and hold a corresponding sponsored visa but would like to take some part-time freelance project-based activities in completely other field (e.g. music industry). Is that allowed? Do I need to apply for the permission and what are the chances to get it? Or is it enough just to inform tax office regarding this side income?

No.
Yes.
Very low; basically you must have a very good reason to want to do that side job. No, wanting to make more money doesn't cut it.