There is no long-distance train departing Malpensa and this airport train is not covered by passes on either day 1 or 7 of your plan. Be sure to choose the shuttle train with the destination of Milano Centrale station (not Cadorna), since that's where you catch most trains to Rome. You can buy the tickets upon arrival for about $100 per person in the cheapest class of service. Tickets booked from home are not usable if you miss the scheduled train.
This is an example of a time when a Eurail Italy Pass offers you both a moderate total price and no worry about missing the booked train. You have to have the pass activated in Milan (probably fastest at the travel agency that's inside the train station or one in the airport) and can buy a seat reservation at the same time. Trips 1 and 2 are each a $100 value at full-fare 2nd class.
Since you can currently get a pass for 6 travel days in a month for $206 per adult in 2nd class or $256 in 1st, I would choose either of those. If there are kids under 12 in the family group, then they can be added to the pass for free. Seat reservations for the 6 trains that require reservations will add $72/€60 per person. Trains between Florence and Pisa don't generally take reservations, so you can just hop on those. The $206 price is based on the current extra-day special and those passes are not refundable or changeable once purchased.