Hi Manos1,
your basic question in your initial post was "What exactly is the procedure?" and
@Kate T pushed you directly to a very detailed KB - articke for this question.
Did you know, that there is as well a complete documentation for Migrations?
Installation, Upgrade, Migration, and Transfer Guide, Parallels Plesk 12.0 ( Plesk online documentation - chapter cPanel )
To 1.) Sorry, there is no 100 % guarantee when you migrate from one software to another. Issues can be caused by all sorts of reason. But Parallels provides as well a frre migration support, if you experience issues, which you can't resolve by yourself:
Further to 1.) Yes, this is possible. It is always a good idea to have a complete backup, but your case should normally don't cause any issues, which might have the need of using it, when you migrate as described with your case.
To 2.) Yes.
To 3.) The process depends on your current software installations and environment, but you should be finished within 30 minutes, if you are an experienced system administrator. We don't know your linux knowledges and we don't know, how fast you get familiar with Plesk, its instructions and the usage. If you would like specific dosumentations, Parallels provides wide-ranged documentations, video tutorials and KB - articles and last but not least, you are able to use the forums for help, suggestions and work-arounds.
During the migration process, there should only be a few minutes of downtime for your current domains, because of a few service restarts.