This will depend on where you work. At private schools they sometimes ask you to write your own and they put their seals on it. At public school you shouldn't really write your own, since the school is from the government.
At public schools they can deny giving you the reference letter if they feel you are not fit to be recommended by their school. This actually happened to me at my old school, due to a hiccup with one of my coteachers. However, by law they are required to give you a certificate of employment which lists your dates of employment. You actually need this for your second or third job in Korea.
Yes! Yes! Be persistent. It's not that they want to ignore you it's that, if you have noticed by now, they don't really have good time management skills or the ability to prioritize. Plus keep in mind that official documents like these, especially at public schools, usually require signatures from several people. If one person is out for a week or day then your coteacher has to wait till they come back.
One thing is if you have recommendations from previous schools or employers back in your home country you will be okay with a letter of rec from your current school. However, make sure you have someone at your school to use as a phone reference as this will certainly be necessary.
Whatever you end up getting make sure you scan it and make a copy. Plus make sure official seals and school addresses and names are on it.
