You are correct. In the old days when people had to use fire for heat and cooking, they would use the bellows to keep the fire going strong. In this context, "pumping bellows" means that the dog was breathing very heavily. Imagine running as fast as you can up a hill, and when you stop you are puffing and panting to get oxygen. You would be breathing like you were pumping bellows. This is a very old expression, and very few people even know it now.

Yes, you are correct. The dog's lungs were moving in and out strongly because it was trying to catch its breath, so it's rib cage looked like bellows.