"They" is completely grammatically correct as singular. In fact, it's been used singular longer than the word "you" has. Ever wonder why we say "you are" as if its a plural instead of "you is"? Because it was plural, but it evolved like language does. "They" is used both as singular and plural, it should be accepted as a translation of "hän"
Yes, that's true, but it has been deemed unwise from a pedagogical perspective to accept the singular "they" as a translation of "hän". One of the contributors has made the following statement about it:
We considered that, but eventually came to the conclusion that too many people would find it confusing. Which is rather sad, since, in many ways, "they" would be the closest that English can manage to a literal translation. :(
Actually it comes from Russian шама́н (šamán), which comes from some Tungusic sources. The English word comes from the same source. And the word is originally šamaani, but later the letter š (which only appears in some newer loanwords) has often been replaced by sh, although this practice is considered incorrect.
So no, it wasn't loaned from English.