Re the free Family Card in Switzerland, I believe you have to purchase at least one other Swiss rail product (Swiss Travel Pass, 30 day Half Fare Card, for instance) to get this.
In Italy, you may find it less expensive and plan ahead to buy nonrefundable tickets for the high speed trains connecting Rome, Florence, Venice, Milan and Luzern. Nobody has any problem buying these types of tickets for air travel, and the same applies to train travel in Europe. You're planning to be in these cities and will no doubt have lodgings booked, so do the same for your travel between them. For instance. I see many trains on May 31 for 88 EUR total for 3 adults and 1 child for Rome to Florence. 112 EUR for Milan to Luzern a week later.
I would just go from Milan to Luzern and stay there for your time in Switzerland. Plenty to do in the area, including mountain top train rides, lake voyages, etc. Assume you are flying home from Zurich, and it is just over an hour by direct train to Zurich airport from Luzern.