When checked, the Synchronization setting for a folder in Thunderbird means an offline copy of your messages on the server is downloaded to your local disk. It has got nothing to do with which messages are visible or not in Thunderbird. With IMAP, you do see the exact same messages on both, the server, and in Thunderbird. This is how IMAP works.

Wrt the Sent folder, check the Account Settings for your Tiscali account: At the top right of the Thunderbird window, click the menu button > Options > Account Settings > Copies & Folders

When sending messages, automatically: Check 'Place a copy in Sent folder'

I thinks that the configuration is ok on Thunderbird, see scren-shot, and the email sent is also on the provider server but if I sent from the provider I don't receive it on Thunderbird. At the same moment I don't see email sent on my mobile or email sent from mobile into server and Thundirbird

if I sent from the provider ...

I'm not sure what that means. Are you talking about webmail?

... I don't receive it on Thunderbird.

See https://support.mozilla.org/kb/cannot-receive-messages

I don't see email sent on my mobile or email sent from mobile into server and Thundirbird

I don't think I understand your point. Can you be a little more specific about what device you're sending from, and to what device? And what exactly the problem is?