I attended a one day class at Le Cordon Bleu and it seemed to be a very nice school. Classes are all in French with English translation with students from all over the world. However I would not trek all the way to France to attend culinary school I would go there to work. You can easily research some restaurants and go and work for a month or two without papers. It is of course better to have papers but it can be difficult to get them.
For instance I worked at a small one Michelin star in Bourg En Bresse and had a great experience. I was there for a little over a month and learned a lot. I think the most important thing is not where you go but if you are ready to go. If you are still not confident in how to make basic sauces, stocks, or how to cook meat and fish properly then you are not ready. Usually you will work for a short stint and you need to be able to pick up a lot in a short amount of time. I also highly recommend you write everything down as you will forget it. I still refer to my notebooks from Italy, France and Switzerland.
I would recommend searching out a local place not a 2-3 star place. Be honest and upfront about what you want to do with the chef. Say that you learned bad habits and you want to become a great cook and you are willing to work hard.
As for living costs, I have been to Paris several times and from an American perspective it seems quite expensive but that is because of the Euro. When I was working in Geneve Switzerland most of the pasty cooks lived in France but commuted to Geneve. The cost of living was much cheaper. I am not really sure I can comment on what the cost of living would be. My guess is that Paris is like any city there are areas that are much cheaper and others that are not. Maybe some of the others here can comment on what areas are economical and safe.
Hope that helps.