I have some projects not yet fully defined; there are only the description of the projects and its sub-projects.
If I copy a project and select the checkbox "Copy sub-projects", I still copy only the "parent" project.

You're right.
Issue recorded as ticket #5144

It would also be useful to be able to define the new customer for the copied project as well.

You can easily change it afterwards throught the "multiple update" feature

Finally, I would like to know what the difference is between "copied" and "registered" status

"Recorded" is the first status when you create an item.
"Copied" is an intermediate status for just copied items. Next status will be "Recorded". This intermediate status is designed to avoid sending emails on creation just after copy. You get the time to change few things (what differs from origin to copy) before move to status "Recorded" and this will send the "on creation" emails.
 

Hi,
For the copy sub-project issue, you must check "copy project structure" to copy sub-project (as sub-projects are in the structure of the main project).
In fact the "copy sub-projects" option shoud be named "also copy sub-projects".