The only essential piece of paper will be a PPL licence:
http://www.ppluk.com/
(Click the 'Music Users' link).
Full-time professional discos usually have public liability insurance but most of the smaller outfits don't. (You probably wouldn't be able to find anyone to insure you at 16 years of age anyway).
If you're simply going to be taking bookings to run discos at parties, etc, then you won't have to worry about child-care regulations. (i.e. you'll only be in charge of playing the records, not running the party). If, however, you intend to hire a hall and run your own events, you would need to check that you didn't commit any breaches of the Children Act. (Your local council can advise on this). In particular, you'd need to ensure that no children under 8 years old were allowed into the event because the regulations are much stricter for very young children.
Chris