How exactly is a foreign name written on a hanko? Is it best to simply use only the first name (in my cousin's case, Rochelle/), only the last name Stout (, I believe?)), or some sort of variation of initials? And if initials are okay, what sort of formatting is acceptable?

A hanko is supposed to be a unique signature, you can style it however you want as long as it's readable. Romaji, katakana or hirakana - but don't make up kanji unless you officially have that kanji registered as your by-name.

Read the part about Jitsu-in: <

> As a rule, you can only register your name as stated on your alien registration card. You can register a stamp of your family name, given name or both. The stamp should print your name in the alphabet, katakana or hiragana. You can register your name in kanji ONLY if you first register the kanji name as a by-name first. This can also be done at the citizens’ service department.