Do any of the AJX pilots here just stay in Japan year round?
Yes....., for the same reasons you mentioned but the company will not sponsor residency cards.
Are you able to bank the money that would otherwise go towards your commuting tickets and hotel accommodations?
Yes....., you will get 2,000 if the C Class is not used.
What about the possibility of applying at ANA mainline if you have a spousal work visa or even become a naturalized citizen of Japan?
NOPE.