You can set up a company if you wish. It will cost probably £200 to buy and you'll have a certain amount of ongoing obligations in regard to it such as filing an annual return for which there is a fee of £15. You'll need at least two directors. If you do set up a company your company can have a trading name which is different from its registed name if you wish, and you can trade under that. If you don't set up a company you would be acting as a sole trader, but again you could have a trading name if you wished for your business, you don't have to do business under your own name. But any contracts will have to be in your own name.
As you probably know a company is a legal person in its own right. But is has limited liability so if someone were to have a complaint about the software, and sue you, it would have to sue the company if the company had sold the software to them. Your own personal assets would (generally) be safe from enforcement of any judgment obtained.
If you have a company you can withdraw any profits (i) by paying yourself a salary, (ii) by paying yourself dividends or (iii) indirectly by selling the company. There are tax consequences to all of these.
As for copyright, your code is copyright without any registration. And things such as the visual design of your program would be copyright. But the actual idea of it will not be copyright. For a lot of "inventions" you can get a patent which allows the inventor to stop anyone else from copying the invention for a certain number of years. In the UK a computer programme is not considered to be an invention; in the US I believe it often is.
Hope that helps.