ehilà
1. no showers, but you can find some in the t3 lounge 2nd floor ariside (after passport control if you come from outside Europe). no idea about the quality.
2. as you're flying ST in business, you should have no problem. bring also the AZ ticket.
3. depends on the time of the day etc. but keep in mind boarding begins 30 minutes in advance normally, so depending on the security/passport check you may want to leave early
4. dolce vita is landside, but t3 and t1 are connected so your choice. it's a quick walk anyway even if you get out of the airport.