"What is a key-money deposit?"
From 300 to 700 US$, depends of the area/city and apt. (smaller cities/towns, do not require key rent deposit).

"So when applying for the position just tell them I want a rent subsidy?"
Depends if school provide rent subsidy. Some schools do not provide. They place the teachers with other teachers in share apts.

"Also, the school will pay for your passport and plane ticket, but not that of your children?"
As far as i know; no school in Korea pays for anyone's passports or their dependants. Schools pay for the visa letter form only(most of the time).