My friend is a Guide de Haute Montagne in the Pyrenees and he always wears trainers up to the glacier when climbing Vignemale (his boots, crampons and rope are hidden in the rocks just below the glacier) saves 1500m of ascent of added weight.
If you are doing the GR 10 then look for Alcool à brûler (from hardware stores and supermarkets) if using a meths burner. The french won't know the spanish name. MSR type gas is widely available especially in the villages in the higher mountains. Camping Gaz is available everywhere as it is a french product and they now do a screw top version as well which would fit the MSR etc. so I would stick with the gas
Are you doing the whole GR10? If so the only tricky bit would be the start at Hendaye as nowhere sells MSR gas. And finding screw top camping gaz could be tricky. However, from the train station at Hendaye you have a 15 minute walk to Irun in spain where there is an outdoor shop in the Mendibil shopping centre (on your right as you walk into town) called Leire Sports and they sell the MSR type gas. So you could get a couple of canisters there before starting off.
Stick to the Spanish side if you are bringing your dog, although lots of spanish people do cross over to the french national park with their dogs and pretend they didn't know. I've seen spanish people camping near Oulettes de Gaube and Wallon with dogs. One thing to note is that a lot of bars/restaurants in spain do not accept dogs indoors, but the terrasses are no problem.
I've been up in the Pyrenees many times with my dog and you have to be very careful with the dogs that are left up there to guard the sheep which is the only time I put him on a lead as they are very protective (they are there to chase away bears and wolves).
Reagrding dogs on french buses it is hit and miss. The bus from Lourdes to Cauterets does not accept dogs, but other lines do.
I've crossed it a few times and have never had good conditions. In fact my wife says she has never seen the dark peak because each time we have been there, there has only been fog rain and snow. But it is a good way to test the old navigation skills. First time was in early spring with snow and ice and I enjoyed it so much the next winter I went to the Cairngorms for a few days.