Since your vet is familiar with the EU Health regulations, perhaps you should ask there first. It's probable that he could direct you to the proper government websites which would apply to you.
You obviously know your dog, but if you have not travelled much with him, you might want to re-think this. If you've never been on a long-haul flight with him, as well as moving around every few days to a different location, you need to be aware of the amount of stress you'll be putting on his system in this short period of time.
I have brought a slightly larger dog from the US to France, and I would never do it again. Even though your dog will be in the cabin with you, you have to think about how you will administer food and water and how he will be able to relieve himself - if he is given a sedative (which is not advised), it's probable he still won't sleep through the flight. He must stay in his carrier under your seat at all times - if your dog is a typical Yorkie, he won't appreciate it.
You must keep your pet in a carrier at all times when travelling by train, bus or Metro and you must purchase a ticket for him if travelling by train - usually around 10 EU. You must handle all your luggage yourself when boarding public transportation or a train, and this will be more complicated with your dog in his carrier. You'll have to walk, feed and water him, too - this won't be easy if you have tight connections.
Most hotels require that pets remain in their carriers if you must leave him alone in the room - this is for insurance purposes. I haven't been to the cities you mentioned, but I live with my dog in Paris, and parks and green spaces are off-limits, as are many stores - especially food stores of any kind. They are allowed in cafes and restaurants, as a general rule. Your dog must remain on a leash when you take him for a walk.
I'm sure you will have a good time, but based on my experience, I really don't think your dog will.
Perhaps you should do some more investigation about the nanny or other boarding options. Your dog will be well-taken care of, will be very glad to see you when you return and won't make you feel guilty.