Franglaise, very sorry to hear that you are going through this. However was brought up by a single mum myself so don't think it is all doom and gloom -- we got through very well, despite the bumpy ride at times!
What I would say tho', and you can shoot me down at will, but as a rule my hard-earned wisdom is:
Kids get on a lot better in single-parent households if they are near the other parent.
Single-parents need to be as near to their network as poss. This helps with all kinds of things, emergency babysitting, company, sense of identity for parent and kids.
Parents, once they decide to separate, need to hang up their boxing gloves and get everything straight for the kids. THis might include going on hols together, living next door to each other, providing access to unpopular parents-in-law etc.
Sorry if this is stuff that you have heard a million times before, but this is what I believe is the case for "successful" splits.
As for France, I have a little experience there too. The benefits system is highly bureaucratic, and often the best person to negotiate on your behalf is a social worker, or local solicitor (notaire) or even doctor. They will get you on the right track and hopefully within 12 months (yes, that long), you will have financial assistance.
France also provides excellent paid-for childcare in cities, in order to attract and retain young families in urban areas. Rural areas aren't so good, and unfortunately local people are often more judgemental there.
Sorry for the long post and boring advice, hopefully some is useful.