Welcome, Naws!
Things like a TEFL cert have NO bearing on your ability to get a visa or permit. A college degree is required in order to get a Residence Permit...and that, not a Z Visa, is what you really want. The Z Visa is just a way to get you into the country and upgrade you directly to a Residence Permit.
It does seem like an age requirement is becoming more common...it was much more rare in the old days (pre-Beijing Olympics). I guess they really want us to have a chance to work after college?
These aren't universal, however. If the job doesn't work out, keep looking. Someone somewhere will be more than happy to get you in.
As for visa/permit rules...as you'll see elsewhere, there really isn't such a thing as a firm, universal set of rules. After considerable experience and research, it's become apparent to me that even the Chinese organizations responsible for issuing and administering the documents don't know any more than we do. Welcome to China.
The reality seems to be that the rules and their enforcement can vary wildly from time to time, place to place, person to person...and then they can all change again suddenly at any time, usually with no warning or notification. Again, welcome to China.
The best bet really is to work with the FAO (Foreign Affairs Officer) or other staffer at your school. With any luck, they should have as good an idea as any what can and cannot be done in their place and time. (Sadly, however, this isn't always true, either.
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Again...if you're stymied in this job, keep trying. Be very upfront about your age etc. Sooner or later, something viable and reasonably attractive will come along.