You can. I live in Washington State and I also sent money through the mail a month ago. But it did take about two weeks for them to receive the money and credit my account.
No. I just sent 20 bucks, my code written on a piece of paper, and I had to use 3 of those forever stamps. But technically, you do have to file the customs form.
I think you just didn't look deep enough. It says "Look up your destination country in the Index of Countries and Localities to make sure you’re following their rules." and the rules for Sweden say
> [...] paper money [...] are prohibited in Priority Mail Express International shipments to Sweden.
So don't pay for Priority Mail Express, just First-Class:
> First-Class Mail International is a generic term for mailpieces that are postcard-size, letter-size, or flat-size and weigh less than 16 ounces [...]
I'm also in WA. Did you just use https://www.usps.com/international/first-class-mail-internat...?
According to https://www.usps.com/international/letters.htm, it sounds like for cash you need to file a customs form. Did you do that too?