Now I probably understand what the problem is.
In JoomSEF's Configuration on the JoomFish tab, set Main language to (none). Otherwise, the main language does not use the /it in URLs.
The /it/home probably works because it is parsed by default Joomla router.

It seems it's working a little bit better.
Please check with our website, by clicking from the Home to the flag (EN and IT)

For both flags we get now:
----------------------------------------------
404 - Category not found

You may not be able to visit this page because of:

an out-of-date bookmark/favourite
a search engine that has an out-of-date listing for this site
a mistyped address
you have no access to this page
The requested resource was not found.
An error has occurred while processing your request.

Please try one of the following pages:

Home Page

If difficulties persist, please contact the System Administrator of this site.
----------------------------------------------

Other parts seems working good.

Only problem still persists on flags only in homepage...

Hi,
I don't see any 404 problem currently on your website, but if I click on IT flag, the webpage is translated to IT and it won't switch back to EN. It seems that cookie is created for IT. You should probably disable the "Save language to cookie" option if you have the "Main language" field set.

Hello,
yes I just tried a lot of configuration last night and I get good result.

I tried also now to do what suggested by you:
Language integration include in path
Always use language? Yes
Translate URLs? Yes
Get language from browser setting? Yes
Save language to cookie? No (was Yes)
Main language: English

JoomFish Router plugin is currently disabled. (was enabled)

Still have the same problem (I also cleared cache)

Hi,
Your website seems to be working fine for me now. Try also disabling the "Get language from browser setting" option, maybe your browser setting is overriding the main language configuration. It shouldn't, so if it works after you disable the option, it's a bug in JoomSEF.