1) Can and should one work in China on an L Visa? Yes and no, respectively.
2) Is this renewing of the L Visa a normal and logical occurrence? Yes, but only for tourists.
3) Could the kids get an X1 Visa? I don't know.
4) What are the odds on ever getting any form of status ( other Visa's ) in China to stay for
good and teach ESL without a B.A. degree? Legally, no chance.
5) Could the kids go to school on L visa's? No, as they're tourists.
6) Can a school really sponsor a none B.A. Degree holder so they can get a work permit? No.
7) If I get a Working Visa ( school sponsor ) can I than bring ( sponsor ) my family on my
own? Yes.
8) What are the odds on getting caught while working on a L Visa? Hard to say.
9) Are you allowed to work under any circumstances on an L Visa? No.
10) If you get caught working on an L Visa, what could possible happen? ( worse and best case
scenario ) Mandatory fine plus deportation plus re-entry ban of at least 3 years. No worse and best case scenario, except for the fine - at least 10,000 RMB and at most, maybe 60,000 RMB, according mainly to how long you've been working illegally.
11) Is it worth the risk ( if any ) to go an work in China as an ESL teacher on an L Visa? I think it's not worth the risk.
12) The difference between a business visa and tourist visa in regards to teaching ESL? No difference - both are illegal for teaching in China.

I should add that Chinese recruiters and agents are mostly liars who want to make money as commission by introducing you to a Chinese workplace, or off of your hard work, respectively. They usually can't be trusted regarding their visa advice. If you're caught teaching illegally, you will be the one in big trouble, not them!