Hi Eva
Driving in Europe is superb with amazing roads and scenery....but it is also quite diverse with different motoring laws, rules and regs applying to each country.
There are some restrictions in place for traveling with dogs and that is mainly to do with Rabies but as long as your pet has the appropriate jabs and paperwork to prove it then it shouldn't present too much of an issue.
A lot of European countries have what they call a 'Vignette' which is similar to road tax for using their road network, those who don't use a Toll system so you pay for what you use.
There is so much information out there about traveling through Europe and it is worth doing the research to ensure you not only comply with all the necessary requirements of each country but also to ensure that you end up on the best route to take in the spectacular scenery.
More than happy to help with any further information you need. Cheers, Trevor