some answers to your questions -
- Registering your business - it depends on whether you will be a limited company or sole trader.
If you are LTD (giving you personal protection against liability/debt), then you must register your business at Companies House. Google companies house .
It costs £15 and can be done online. Select "limited by shares" and "adopt model articles" where required.
*** note - if applying to register, ensure you are on the Dot-Gov.uk site- there are fee charging sites as well*** You also need to register for self assessment (same as a sole trader , below)
If a sole trader, you register for self assessment as soon as you invoice for work done (no need before that). https://www.gov.uk/working-for-yourself/what-you-need-to-do . It is a 2 stage process, you get a Unique Tax Reference - UTR within 28 days, then apply to submit online tax returns and get a second access code. The tax office give a lot of online support for self assessment.
Re invoicing , as long as you are in the EU, you can invoice fairly easily. You have to abide by the tax law of the country where the transaction takes place, so you must follow the VAT law of the UK (if you turnover more than £82K pa, you MUST register for VAT. Below that, it is voluntary - a lot of businesses prefer VAT registration , esp if your clients will be larger businesses.
Your name - must be clear on invoices and legal documents. Your legal name must be on your website in some format.
You can call yourself anything you like, provided that it isnt subject to trademark (eg Alphabet, Amazon) or a created word, subject to copyright (primark, easyjet) .
You can search for trademarks at www.ipo dot gov dot uk (cant post 3 links!)
If your name carried value and you want to ensure no one takes it, then you can apply to protect it via the IPO website (£180 for 1 category, additional fees for other sectors to be protected). Some words cant be protected if common in the sector (eg design, graphic, tech, etc). You dont have to say what you do, but you cannot mislead and suggest a status that you dont have (eg Chartered Institute of..., Royal Society of....Bank of..... etc
If you are just using your own name, then its not necessary to protect it, if no value , but if it starts to carry weight (eg like Saatchi &Saatchi ), then you can trade mark it.
With anything like that, id suggest some legal advice, but from what you say, i dont think it would be necessary.