you don't need to register the business, just trade under you own name for now.
keep detailed records of any money you make/spend.
i know a few people who have set up businesses at your age, doing more or less the same thing.
keep your job, and declare yourself employed for things is the best way to go. (Car insurance i believe is cheaper if your employed rather than self employed for example).
it seems unlikely you will earn any taxable amount so don't worry about that. (Although keep your records anyway).
just keep your personal bank account for the transactions, its ok as you trading under your own name anyway. also business accounts may not be available for people your age.
if you want you could open a second current account just for the business but again under your own name.
when it comes to invoicing your clients you should research some suitable templates for this to send to your clients.
there is a clearly defined procedure for payment and transactions that you should look into, especially if you are dealing with businesses. (It involves credit notes, days to pay etc etc).
if they are personal clients that you work closely with then you could ask for cash in hand or a cheque to avoid all the invoicing procedure.
contrary to popular belief you do not need to register your business with anything unless you are on the stock market or declare limited liability (Which you can't do in this case).
in the eyes of the law you and the business are the same entity.
to sum up, you are trading under you own name, and even if you weren't it doesn't matter. Just go for it and make a start.
But remember that while any money you make is yours so is any debt you get yourself into.
oh and Good luck