Vanessa, before I attempt to answer any of your questions I need to know if you hold an EU country passport. The regs are very different if you do or do not.
One thing I would say is that your chances of finding a job, in this area, anyway are practically nil. Many newcomers have fallen down at this hurdle; thinking they will just be able to pick up a nice little part-time job. Unemployment is very high in France at the moment and they, quite rightly in my opinion, give their own citizens preference when recruiting. You would also need to be fluent in French to be considered. Perhaps you are, but it's something to bear in mind. Likewise your children will find it hard going at school if they don't have at least a grounding in French; it is quite usual for English kids to be put back at least a year unless they are very young when they seem to pick it up extremely quickly.
We have lived here for nearly ten years and are very happy but we didn't arrive needing to earn a living from a local job. We are both self-employed in computer based work which could be done from anywhere in the world.
Tempy