I also use Mullvad and I pay by sending an anonymous envelope of cash to Sweden with a random number scribbled on a bit of paper. So far all of the cash I’ve sent has been added to my accounts. There’s obviously no way to know whether they follow through on their no-log claim but my sense is they’re pretty trustworthy. At the very least the person who opens their mail is.
I do indeed! I’ve sent USD, GBP, and Euros from all over the world - Europe, North Africa, South America, and the US. (I was a “digital nomad” pre-pandemic.) They even give you a one-time-use token for cash payments so you can maintain the anonymity of your account number if the letter is intercepted.
Other than the letter I sent from Morocco, I never included a return address.
That link automatically creates a new account. No names, no email address (except if you set up a PayPal subscription), not even a password. Your account number is literally all you need. They even have a FAQ that provides a generalized database schema.