Your 12 month teaching period starts as soon as you get your alien card and can last until the expiry date shown on that alien card.
Employment contracts with Native English teachers are worthless and cannot be enforced by the Labor office, so if you are determined and you are dealing with a hagwon, the hagwon will also have a hard time enforcing it.
The contract is simply proof to Korean Immigration that the school has a job for you, hence your working Visa. The school can however cancel your working visa at any time.
Quick money seems the only good reason to work in Korea. My advice is avoid working there if you can. Otherwise make friends with honest Koreans who can help you deal with the Administration or with hospital because you will need them.