1. Yes, you absolutely have to be a native English speaker for the government driven EPIK programme. It's also the same for virtually all private academies these days. Nothing else matters.
2. As a student, you will only be able to find part time jobs which don't pay much (less than $5/hour)
3. You absolutely need TOPIK5 for studying in Korea if you apply directly which isn't easy at all. It would be easier and less of a financial burden if applied as an exchange student instead. Go to your local university first then look for your opportunities.