I assume that by "FEC" you mean "Foreign Experts Certificate"? If so, as far as you're concerned it's mainly just a precursor to a Residence Permit. It has nothing to do with getting paid.

What does your contract say? Some contracts do cover a full calendar year, but contain provisions specifying that Summer (and sometimes other) holidays will not be paid. Anything about this in the details of your contract?

This is almost always done by the employer...I think it ranges from difficult to impossible to get one of these on your own and without a specific job lined up.

The employer provides the local Labor Bureau some proof that you have desirable knowledge or abilities- such as a university diploma, a teaching certificate, certification in a specific skill, etc.- plus copies of your travel documents and of course good old-fashioned mountains of Chinese paperwork.

Naps are taken, tea is consumed...and a few weeks later a small brown book is excreted that you can then use to go obtain your Residence Permit.

This, by the way, is why schools are so insistent upon a university diploma. It has little to do with on-the-job qualifications and much to do with getting your FEC.