Is it best to simply use only the first name (in my cousin's case, Rochelle/ロシイル), only the last name Stout (スタウト, 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: http://www.city.kakamigahara.lg.jp/international/english/3101/003106.html
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.
My experience as an American living in Japan is that you are not expected to use hanko. My signature in the hanko space on documents (bank and other) was regularly accepted. (Yes, I did have to kind of squish my signature in, almost like initializing instead of signing.) I have lived and worked in Japan multiple times over the years, but have never had long-term resident status or citizenship, so your situation may be different. But if you ask, you may not have a need for a hanko.