In the US, anyone can sell good or services and do business without formally registering as a business. You're automatically classified as a sole proprietorship, with your legal name as the business name. But there's no liability protection, since you and the business are the same legal entity. That's where LLC or incorporating comes in, along with lots of other reasons to want to formalize the business as a legal entity.

If your turnover is low enough you don’t need to be VAT registered in the UK. Also, in the UK you can be VAT registered and be a sole trader.

I think you just take the cash using Stripe or whatever. It gets deposited in your bank account and you claim income tax on it.