Thanks for posting a new thread. I assume you've already bought your airline tickets. If not, it would be better to fly into Venice and out of Rome on a multi-city ticket.
Is May 12 the date you leave home or the date you arrive in Italy? What is your departure date?
Since your trip is short, do everything you can to minimize hotel changes. They are time-consuming and really annoying, especially when you're trying to corral eight people! I'd put all the Rome time at the end of the trip since you need to return there for your flight home anyway.
The arrival day from the US is normally experienced in a jet-lagged fog (with some people having a much harder time than others, another issue for a group of eight), and most of us don't really count it as a day in Europe. One very good use of that day is to head to one of the other cities you plan to visit. If you fall asleep on the train, so what? That gets one of the necessary travel days out of the way right away, when you're going to be rather unproductive anyway.
Many people will suggest heading for Venice first thing since it's a good place to deal with jet-lag and is the destination farthest from Rome. The tricky part is knowing what to do about the train tickets. Since there are eight of you, you'll want to be prudent about their cost. Buying tickets way in advance can save a lot of money, compared to waiting until your arrival. But the lowest prices are on non-changeable/non-refundable tickets, and it's difficult to predict how long it will take you to be ready to hop on a train. Your flight could be late, there could be a back-up at the Immigration desk, there could be a delay in receiving your luggage.
You can explore train schedules and fares on the TrenItalia website. (Do not buy a railpass! Do not buy overpriced tickets from RailEurope!) You'll need to use the Italian spelling of the city names/train stations: Fiumicino Aeroporto, Roma (Tutti Le Stazione), Firenze S M Novella, Venezia Santa Lucia, Napoli, Siena.
It appears that if you wait until arrival and buy your tickets at the airport train station, you will pay 90-130 euros per person (I assume less for children) for a ticket from Fiumicino Airport to Venice. That's a lot of money. When I look at prices for early March (buying about 90 days ahead of time), I see fares mostly between 30 and 50 euros. This is why lots of travelers pin down their travel days very early, to snap up those cheap tickets. But it's tricky to do on the day you arrive from the US, as I explained above. Some travelers just pick a departure time 3 or 4 hours after their flight is due to arrive and cross their fingers. I confess that I am not that gutsy.
If you're not comfortable gambling on non-refundable tickets to Venice, you might instead start your trip in Florence. You could wait until your arrival and pay last-minute prices for train tickets to Florence. For today, that might be as low as 50-60 euros per adult, but it could be higher than that.
I don't think you have time to visit both Venice and Naples/Positano. All of the times you've proposed for cities are really too short. When you spend two nights in a city, you have only one day plus a few hours there. You can't see a city like Florence and take a day-trip in that amount of time. It will be a better vacation if you stick to Rome, Florence (with a Tuscan day-trip or two) and either Venice or Naples/Positano. I'd go with Venice because it's easier to reach and may need fewer days, which frees up time for nice visits to Florence/Tuscany and Rome. Most folks who go down to the Naples area want to visit the Amalfi Drive, Pompeii, maybe Capri and the archaeological museum. That's a lot more moving around than you need to do in Venice, though there are some nice side-trips you can make from Venice if you have time.
Are you, by chance, planning to rent cars?