Aellis,
First of all, this is France and not Spain, but I am sure you know that already.
The way from Le Puy, is strictly speaking not a pilgrim 's way, but it is walked a lot, and increasingly the last few years. The "Chemin du Puy" follows the GR 65, one of the many excellent long distance hiking trails in France, which are indicated by white(above)/red markings .
To answer your questions:
1. The cheapest ,and present every 10 to 15km or so,are the "gites d'etappes", which can range from 7 euro to 15 euro/night. These are all more luxurious (provide sheets etc) than any of the refugio's in Spain. They are usulally small, so booking in advance is essential (certainly in the June to September time frame). You have to do that yourselves, and one idea is to think and book a few days ahead.
There are a few, not many, refuges, which are usually connected to churches. After that you will still have the chambres d'hotes and hotels, which can vary in price considerably (from cheap to outrageous)
2. You can generally walk the same distances as in Spain, possibly a bit less, because the trails can be a bit more challenging.. the markings also are not always clear. The GR 65 is walked a lot less than the Camino, so depending on the time of year you plan, outside the summer,you might meet a lot less people on the trail. Many parts become unpassable from November onwards.
3. Some gites have kitchens, some don't.
This being France, some knowledge of french would be very useful, since along the rural areas around GR 65, english is hardly spoken, and spanish not at all. Shops are generally open until 1.00pm and then again after 4.30pm. Meal times are quite different from Spain, lunch from 12.00 to 2.00pm and dinner will start after 7.00 or 8.00pm. But you probably know all this.
The whole route is described with little excellent maps and desctriptions, including adress and phone numbers of the gites in three topo guides, which are in french, but are so clear that most people will be able to understand them.: " Sentier vers Saint Jacques de Compostelle, ref. 651, du Puy a Figeac; ref.652, de Figeac a Moissac; ref. 653, de Moissac a Roncevaux " from the Federation Francaise de la Randonnee Pedestre",
http://www.ffrp.asso.fr
These thin excellent guides/maps beat everything I have seen elsewhere. The French certainly can make maps better than anyone else,
Hope this lengthy reply helps,
Kees