Do they expect some kind of standardized form of payment, such as an "invoice" in a
specific format, which I have to e-mail them as a kind of official receipt?
No. You need to learn the english terms - or how to use a dictionary to look them up.
An INVOICE is a statement of what you HAVE to pay.
A RECEIPT is a statement confirming something HAS BEEN PAID.
So, per definition an invoice can not be a receipt. An invoice can ALSO be a receipt if you mark the invoice with a field "paid in cash" with a date and signature from you.
Or would they be perfectly okay with just being given my standard IBAN/SWIFT/whatever
numbers as well as my personal name and the amount?
An email with IBAN/SWIFT/whatever and the amount to be paid IS CALLED AN INVOICE.
Now, whether it is itemized or not - I have written a ton of invoices in my life where it says "item 1: amount X" and that was it - mostly in combination with a SEPARATE item tracking i.e. the amount of hours worked.
Also: "companies" -you would expect to know that with hundreds of thousands of companies around, there will be all kinds of different behavior.
If they demand an invoice, does that require that I have an actual registered company?
No, it does not. THAT SAID: If you do a lot of it without a registered company, you may want to check out local prisons first because you will possibly get into all kinds of trouble.
I'm trying to determine whether or not there is any point in me working on a service for
companies or not.
You mean outside of EARNING MONEY? You think the only way it makes sense to work is if you can betray the government by not following the regulations? Because otherwise let me in on some news for you: Pretty much every business in the world does so because they think it makes sense to work for clients to earn money to pay their bills.
I've already determined that I cannot charge money from "normal people" because they use
PayPal, pay cards (Stripe) and other payment services from which I'm locked out. But
maybe companies are different in how they pay, and will do it like this, as an
international bank transfer straight to me?
Dude, why would any company bother with an international bank transfer for a small amount of money? Also why would you be locked out of a service like paypal - outside of (a) living in an area that is blocked, then no company will deal with you also or (b) having been doing some really shady stuff and then your problem is more like you shady stuff.
Companies generally do not like dealing with shady people. They also VERY OFTEN do not like to deal with unknown factory, insisting on general written contracts on a lot of stuff. Because they are companies with workflows and different departments. You do not follow their requirements? Then either you make something they NEED that is not replaceable (which is unlikely) or they - do not do business with you.
And IF they do business with you, you better do not appear to be the kind of clueless guy that tries to figure out what is an invoice. That behavior generally raises so many red flags, it gets your account banned from paypal etc. because IT IS SHADY.