1. Part-time work is always possible, though with limited French it may not be easy to find. In France there isn't much of a distinction between full-time and part-time work - at least in terms of social insurance and such. You'll still need to deal with social security numbers and social insurances - and registration for VAT, if you go the freelance route with your work. It may wind up being more hassle than it's worth. But see what the job market it like when you get here.

2. Most communities have "halte garderie" services available. Ask at the mairie (town hall) when you get here. It's usually a subsidized service of the local government, and much less expensive than private child minding services. Many families sign up for just one or two days a week on a regular basis and the kids seem to have a good time. Most foreign kids wind up speaking fluent French after only a few months.

3. Lyon is the second city in France. Beziers is much smaller and (I suspect) a bit more laid back. If you're interested in being near the mountains, you might want to consider the area outside Lyon - someplace like Annecy, which is definitely in the mountains, or Grenoble. Then again, I'm not keen on living smack in a big city - you may prefer that.
Cheers,
Bev