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.
Oh and forgot to say San Sebastian is an excellent choice. Stay the night, experience the Pintxos (tapas) and for foodies it is a pure delight. A wonderful town with a fantastic atmosphere. Whenever my wife and I fancy a night out that is where we go.
The only place to eat if necessary is called "Café Leffe" on the main street nearly opposite McDonald's. As Pedro says plenty of places for food but I know all of their suppliers and what they offer, as they cater to people on pilgrimage who are not really interested in quality food. And if you're staying in Gavarnie for a night it will be quite quiet for the evening. It is very popular but most people are daytrippers from Lourdes and other towns. So after 6pm it will be quite relaxed.
Have a good trip. Don't stay in Lourdes too long, and if you are a foodie avoid eating in a restaurant there. They are all disappointing. Most restaurants use the cheapest sources to provide meals very cheap for the Pilgrims.