Every country in Europe is different when it comes to PhDs. What exist at most is the recognition of a PhD delivered in one European country as nearly equivalent to a PhD delivered in another.
France, Germany, Italy, Poland have more or less different notions of PhD and a huge lot of facilities delivering PhD (in France, be aware of our Grandes Écoles: some of them, like École Polytechnique or ENS, deliver PhDs). French CIFRE is pretty unique, even if of course you have near equivalent in other European countries.
I've got a French PhD in AI (in 1990). The formal title is Docteur de l'Université. Stricto sensu there is no PhD in France.
Should I think first a viable project, or should I stay flexible? Should I just look online for some position in the research fields I like most, or should I maybe focus my attention on professors, institutes, or should I even select the country first? Should I rely on the contacts of my bachelor and master universities professors, or should I try by my own means?
A PhD is mostly a human adventure. Not all PhDs get a job in academia (after it). Not all PhD graduates are paid for their PhD. A good PhD depends upon the people you are doing it (more than the institutions you are doing it).
You forgot to explain the most important.
why do you want to start a PhD? In what scientific domain?
In France, rules are different for a PhD in philosophy of science (or in medieval history) and for a PhD in computer science: in practice, you are required to get funded in computer science, but not in philosophy of science. I guess the rules are also different in Italy or in Germany (even the formal title is different).
The german Doktoringenieur title has no exact equivalent in France. Of course you do find near-equivalent.
The European research funding programme like H2020 have hard time defining what approximately is an European PhD. In practice they are funding many PhDs.
My opinion is that you should look for PhD advisors (so persons and their scientific work and reputation, not institutions).