I want to be able to add a press pack download to http://www.aliaskid.co.uk and I am wondering if it is a good idea to embed an MP3 of a song into a PDF file using indesign or whether this is even possible.
Yes, you can embed an MP3 file in an interactive PDF in InDesign.
Is it a good idea? That I can't answer. I am unfamiliar with what you mean by a "press pack download" and the website you're using.
The problem with including multimedia in a PDF is that there are many PDF readers, some incapable of properly playing multimedia. I'd suggest adding a recommendation that viewers of the PDF use Adobe Reader or Acrobat on Mac or Windows for multimedia. Most PDF readers on an iPad or Android tablet won't handle it correctly.