With 5 days in Tuscany, I'd recommend finding a central B&B in a farmhouse or villa and doing day trips from that base, in a different direction each day. Car rental is highly recommended in this case, but whether you rent from Rome and drive up really will depend on where you end up staying so I'd say to think about that after you've booked your accommodation.
So as for where to stay, I'd say Chianti or slightly to the south of Siena as being pretty central to reach all of the places you mention. Take a look at the apartments on this page dedicated to
Chianti and on this one on
Val d'Orcia.
Most apartments rent by the week, but if you see some you like, you can still contact them and see whether early June has availability for shorter stays. In particular, take a look at
Casolare di Libbiano and
Palazzo Malaspina, which are both really central in the San Gimignano/Castellina area.
Where do you leave to Paris from: Rome or Pisa or Florence? That will really affect whether you can fit in Cinque Terre either at the beginning or end of your trip. If you have to head back to Rome, I'd cut a day from Rome when you arrive and head to CT first, then head back down into Tuscany to then head back to Rome for departure..... if you leave from Pisa or Florence, you could do it at the end, giving yourselves a full day in CT. If you're traveling by train to Paris, then you could leave CT to the end of depart from CT directly, passing through Milan (take the night train from Milan).