MULTILOCATIONS
I changed your settings to NOT use neighborhoods (because you did not have any neighborhood entered anywhere).
I also changed the setting to use default country, instead of selected countries with one country only.
It all seemed to work fine for me now, until …
CATEGORIES
I think you have might have renamed your category “Restaurant” to “Ristorante” at one stages, and for those listings, you get the error that the category is missing (not the address).
Just delete the category and re-add and it works.
GD BOOSTER
My test listing had no problem with the address from the frontend, but when I wanted to see the published result I got this error:
Fatal error: Invalid opcode 153/1/8. in /home/baltazar/public_html/geodirectory/wp-content/plugins/geodir_gd_booster/includes/share.php on line 420
So I disabled the booster until Stiofan can look at that.