Your questions have largely been answered by Turnoi and Kevin so I won't spend a long time answering them.

Simply, people do work on L, F an M visas but it as we all know is illegal and not advised. You would feel like a fugitive and be always looking over your shoulder. You have to lie because you might accidentally mention to a local person where you work, then they will tell the authorities and then you will get a visit and they will pick up a reward. I would not want to live like that and doubt you would. If you were caught it would be you in the firing line not any agent or employer.

The fact you are intent on bringing your family into the equation opens up a whole different can of worms. The visa regulations even for those legitimately here are a labyrinth never mind if you are here without a legal status.

Employers who can offer z visas usually will but the reason why there are far fewer compared to those who aren't is because to get legal status in China to be able to employ foreigners requires a vast amount of legwork, financial status and other things which generally new or small training centers simply have not got or are able to get. During the summer I was approached by a new training center with an invitation to see if we would be able to have their intended foreign teachers listed under our licence. When I advised them that our licence is for a set number of foreigners which we have and even if it was not we wouldn't entertain this no matter what the financial incentive they lost interest but this kind of thing does go on (or rather did go on as it is becoming almost impossible to do now).

It is getting more and more difficult even in the remote and less desirable provinces to circumvent regulations and the widespread changes that took place just a few months ago have pretty much put paid to that element of things. I have mentioned my feelings before about this already in previous threads.

There were and possibly still are cases where employers will generate a degree and submit it if it is a case of they can legally obtain z visas but the applicant does not have a degree. This was largely overlooked but now because of the countrywide live system is becoming almost impossible. As an example in my province we can submit copies of the documents online to get initial clearance but we must supply original copies for verification at the time we go to physically obtain the Foreign Experts Book. I have even had to change our contracts to state that if we employ you and you cannot later supply the original of any document then we will have to terminate you immediately. It has gotten that strict.

I pity to your situation as I am sure you would be able to make a go of it and if I were you I would be looking at international schools, but sadly because of the degree situation it is going to be a tough challenge for you. The rules specify at least a three year verifiable bachelors degree to be able to qualify for a Foreign Experts Certificate, z visa and resident permit. Regarding X1 and X2 visas it is not something I have had experience with so can't honestly say.

An agent which has rather a disagreeable reputation not to far from where I am started getting foreigners X2 visas because they offer a six month stay without the need for a resident permit but I heard that that got knocked on the head this year.

Bringing me to my final point regarding agents and recruiters, not all are out to deceive you but sadly a lot only care about their commissions and rather like a hotel cares about 'bums on beds', they care about 'a body in front of a class' and will do all they can to achieve that, often promising you the world but when in fact you get to China the reality is vastly different and everything was a 'misunderstanding'. Our school stopped using recruiters and agencies a long time ago and the problems we had when we did were almost eliminated overnight largely for this reason, the web of lies they used to tell applicants. I appreciate some people with no experience will not know where to look but you can with a bit of detective work find your own jobs without their help, it just takes more time and effort but it is achievable.

Good luck whatever choice you decide to make.