I did a small task (writing an article) that requires one-time payment. I am not a registered company/freelancer, but the person asked for an invoice. Do I have the right to issue it? What should I do after that? The client lives in Canada, I'm from Europe.

There are templates in Excel and many of the accounting programs. Just take an invoice and copy it and fill it out. You don't have to be a company to send a bill...which is what an invoice is. Be sure to give it a number and you don't' have to start at 1. Yes a letter will do stating all the facts.

Anyone can write out an invoice (legally).

There are plenty of templates on Google docs. keep a copy for tax purposes.

Yes

I have freelanced locally using a simple form and My Name, Consultant as the business.

just a letter with terms and price is sufficient