Teaching Job

1 & 2. You need to apply for a work visa to work in China legally. Before that, your employer need to obtain a Work Permit for you from related China government (they should know which government and where to do that as an international school with foreign teachers).
3. A single-entry costs USD50.
4. After all the requirements are submitted, it usually takes 4 working days to process it. Usually, one should start to apply for the visa 1-3 months before departure.

Thank you ma'am.
5. My fifth and last question please.
I read the passport must be valid for 6 months before you can apply for a visa to China. Does that mean, a newly acquired passport under a month wont guarantee you a visa?

Yes, you are right, the remained validity of your passport should be at least 6 months.

The validity of your newly acquired passport should be several years, so it should not be a problem for you.