Some Japanese websites have no problem addressing me as Amandaさん, but how common is this in human-produced Japanese? Under what circumstances would someone refer to someone else as Amandaさん, בועזくん, or Екатери́наちゃん instead of アマンダさん, ボアズくん, or エカテリーナちゃん?
It's used quite normally. My business emails, spam mail, post from the bank or government are all normally addressed to
David 様, or whatever exact name they happen to know me by. The same goes for more informal communication using 〜さん or other less honorable honorifics. A certain client of mine is addressing me as
DAVID SAN in emails. Non-roman, non-Japanese alphabets probably won't be used that easily though, mostly because few people can understand or write them. If you introduce yourself to people using the katakana spelling of your name, people may be more likely to use it instead of rōmaji. It always depends on the individual author though. Some will try to find out how to spell your name in kana, others may be glad for the opportunity to use rōmaji.
I can only speak to my personal experience, but my Japanese coworkers usually refer to us as [English First Name]さん (like Michael さん), even though we sometimes sign our emails in Katakana.
Perhaps it's because the Katakana isn't always exact (in fact, some names sound completely different in Katakana because the syllables don't exist in Japanese). By using the original language, they can write your name as it was meant to be written instead of a foreign adaption of your name.