Hey Csaba,
regarding your questions:
1). I’m note sure I understood exactly what you would like to achieve (the individual project template already displays the project title); could you please explain further? In any case, if you want to customize the appearance of the individual project page, the template file to modify is the “single-project.php” file.
3). The project type icons do show up on portfolio tag pages, e.g. here in the demo. Could you please clarify?
Sincerely,
Andrii / Satori Studio