Not only a bad idea on so many levels, but not likely possible. Different countries have different regulations about bringing animals in. Some of the "no-rabies" areas like the UK, Ireland and Scandinavia are extremely strict.

To travel with a dog around the EU, it first needs to be microchipped and given a rabies vaccination. The minimum waiting period after the vaccination is 21 days before you can travel to another country. The strict countries require a blood test and a minimum 6-month (it took a year for my pets) waiting period before you can enter. The UK will also only allow animals to enter through approved routes, which can be incredibly expensive.

Before travelling to a new country, the dog has to be treated for worms and other parasites, within a few days of travel. Some countries also require another vet treatment upon arrival. This means up to 2 vet visits per country you're going to.

Pets aren't allowed at any hostel I know of, usually only large hotel chains. Accessibility to businesses varies from country to country, but again, animals aren't allowed most places. Some transportation won't allow animals at all, or you may have to pay extra, possibly even a full-price ticket.

You also can't just register a non-trained puppy as a service animal. Unless you have actual disabilities that would be helped by such an animal, I don't think that aspect of the plan has any chance of working. On the subject of training, a puppy needs a stable environment in order to learn good habits. Moving an animal constantly is extremely stressful and confusing for it, especially one so young and unfamiliar with you.

And you can't just feed a dog whatever you eat, even if you eat healthily for a human. Dogs have different dietary needs, and a raw diet must be carefully planned. I don't see how this is possible while backpacking.

I'm a former veterinary technician and animal rescuer, and I highly advise against attempting such a trip with an animal. It's really not fair to the dog, and you should put its health/welfare in priority.