I am an American; if I were going to live in Japan for any amount of time and conduct business, on official documents, could/would I write my name as Americans do (first-last) or as Japanese do (last-first)?
In most of the time you're going to keep given name first, unless otherwise designated (like in some application forms). Japanese know, and expect that Western names (actually, almost everywhere outside East Asia) are expressed in this order.
However, you may see your name spelled with family name first in some documents, when they are compiled for indexing purposes (e.g. employee list). It doesn't mean you'd be addressed in that way, though. If you're asked to tell your name always from the family name, I'm sure that it's a very special practice in that industry.
Even if you are writing your name in katakana, you would do it in the first-last order. For example, Donald Trump would write his name as ドナルド・トランプ.
This depends on whether you are writing it in Japanese or English, I would say. It is common even for Japanese people to do
first-last when writing their names in English.