I don't even know what I can offer you for your needs. But it seems to me that a vacation in northern Italy on tour https://www.freetour.com/florence will suit you.
There are a lot of interesting places that you might like, given that you have a dog. The southern part of the country has more water, such as reservoirs, fountains, and even the sea (you probably know).
The northern part is suitable for active sports, games, and recreation with the family (gentle option without the extreme). But I do not know if hotels can accommodate dogs, perhaps you need to rent a villa or a small house.