Hi - Im an accountant who was previously a web designer (which paid my study fees)
If you have set up the business yourself, this makes you a sole trader and you wouldn't need to register your name with Companies House www.companieshouse.gov.uk unless you were a limited company with more than one business owner.
If you are a sole trader then you can just call your business whatever you like so long as it doesn't infringe trademarks ( www.patent.gov.uk for trademark search ) You register self emplyed with Revenue & customs www.hmrc.gov.uk.
You can buy a domain name really cheap nowadays which comes with free hosting space to place your completed website. I used www.nameroute.co.uk as its really simple and cheap. You upload the website using ftp programs like a windows explorer type way of transferring files from your pc to the domain itself.
A website is actually really easy to do yourself with a lot of practise. If you have MS Frontpage installed, & an image editing program you can do it yourself if you think of each stage as a jigsaw. Other than that, do a google search for cheap webdesign. take on a freelance webdesigner who will be much cheaper than a company as the freelancer wont have the same overheads to incorporate into the cost you pay.
It may be worth registering with revenue & customs for VAT, if you go for a flat rate scheme you charge 17.5% VAT on every sale which the customer pays, but with flat rate scheme for your industry you pay a percentage of e.g 11% meaning you keep the rest and for accounting purposes treat it as misc' income on self assessment.
Good Luck