Sounds good!
Regarding onward tickets, that is more about the airline than immigration. Many airlines are strict on the requirement that a passenger has a return ticket or onward ticket out of the Philippines within 30 days. Many don't know about Balikbayan privileges. Some have notes in their system but a supervisor has to been called in to find and approve the waiver.
I assume you will have one and your wife does not need one, but for the kids? You will all have one way tickets? There are actually "rent a ticket" services where for 10-12 USD they will loan you an onward ticket. Or, you can seek out a cheap "throwaway" ticket on one of the Philippine airlines. You might find some cheap ones on sale. Even if you are not asked for it, it is cheap insurance.
Last resort, if you don't have one and they insist on it, you can buy one on Expedia, show it on your phone, and refund it within 24 hours.
Thank you for your service! Enjoy your time in Korea and the Philippines! No chance for your family to be with you in Korea?