Hi Christophe and welcome from the "Montfort-sur-Meu 3rd-age university", Brittany, France!
1.- Why didn't you post your question on the French-speaking forum?
2.- "While trying to restore it, no error message but the HP5 activities aren't restored at all basically. I suppose it is due to the absence of H5P plugin (?)."
That's pretty obvious! How do you expect Moodle to restore activities that have been created with a contributed plugin to a course where that plugin has not been installed?🙃
3.- I have successfully installed that Italian language course backup on my local Moodle site where the H5P contributed plugin is installed, no problem. On the other hand, when I tried to install that backup on another local Moodle site without the contributed H5P plugin, I did get quite a few error messages (probably you did not get those error messages because you had not enabled the development debug messages on your Moodle site).

4.- You ask "can the H5P plugin (from H5P) live together with the Moodle H5P core integration" The answer is Yes, of course, but you must understand that the H5P contents created via the H5P contributed plugin on the one hand and those created in the Contents Bank on the other hand are not interchangeable.
Unfortunately there is no way to "bulk" convert H5P contents created with the contributed H5P plugin to H5P contents inside a Moodle course's Contents Bank. It can only be done one by one.☹️