Rudelstein,
Living in Seoul not officially sanctioned and may have Korean Tax impications. Any accomodation would be at your expense.
Your family get 4 firm ( and that is real firm) business class tickets per year. Your current wife, if second time round, and all your kids qualify and can travel independently if they wish. Each time you pass through Seoul counts as one ticket.
SYD - SEL - SYD is one ticket.
SYD - SEL - LAX - SEL - SYD is 2 tickets.
System works well and is one of the plus points of the job.
Regards