The only inclusive service you’d get would be on Florence to Rome, where you get a coffee / Prosecco & a snack in First on the Frecciarossa.
On the regional trains between Nice- Monterosso- Florence, there’s usually no First Class at all.
On the TGV in France, just expect a more roomy carriage, 2+1 seating.
You should compare passes vs regular tickets. With Eurail you'll need a 10-20€ reservation for Paris - Nice and 13€ for Florence - Rome (extra costs).
Get tickets for Paris - Nice in advance as prices increase closer to travel date and this train could definitely sell out.
Fixed price for Nice - Florence : it doesn't matter when you get the ticket.
Florence - Rome : availability is never an issue, even on the day. There are 3-4 high-speed trains an hour.
I'd go with 1st class for the high-speed journeys as these tickets usually cost 5-10€ extra, worth the little upgrade.