I resolved all my issues with the help of the guys of Intrepidlogic that made Calibre Cloud.
When I installed the standard versione, that works like Leger Calibre, I relized that it reads the db from Dropbox rather that directly from the device Calibre folder (and this was for me a bit frustrating since each time you open a document from this app, it is downloaded in a new copy.
I contacted the developer asking him if there was any chance to get the feature I was asking (that is the possibility to read and access the calibre db directly on the device and keep the calibre db syncronized with dropsync pro).
The developer (named Jeff) found my suggestion helpufull and promised that a new realese with the requested feature would have been released soon.
Yesterday I received an email from this nice guy from Intrepidlogic where I was informed of the release of the new version of Calibre Cloud pro.
I bought immediately the app and tested it.
Everything works perfeclly.
The app reads the Calibre db stored on the device (alternatively you can choose the cloud solution that reads the library from Dropbox or Google Drive).
I think that these guys from Intrepidlogic deserve a big and loud clap for the quick support and the attention they paid on my issue.
I have also to thank you a lot for the support you gave me