In (4), though, I do not get the French exception. There are many countries in Europe with a president and a prime minister
In most of those countries, the role of the president is mostly ceremonial and/or has to do with foreign policy outside of EU.
Germany and Italy are exceptions, not the rule
22 countries: Prime minister
5 countries: President
https://en.wikipedia.org/wiki/European_Council#Members
Yes, but what does it have to do with the original point about countries with real power presidents?
In most of those countries, the role of the president is mostly ceremonial and/or has to do with foreign policy outside of EU.