How to copy a project suite and all the associated projects folders with all its contents?
This section of documentation may be useful. I suggest starting with the Managing Projects, Project Suites and Project Items section
The short answer is that the folders you see in the TestComplete UI are not actual Windows folders. They will not automatically line up, even if you name them the same things.
See what you like about the options in the documentation and then let us know if you have more questions.