You have not specified if you will have a car or not.
Pisa: since you have no choice but stay 3 nights for business reasons, maybe, besides Lucca, you might consider another day trip to the Cinque Terre. About 90 min away by train. Pisa itself has nothing to offer but a piazza with a cathedral, baptistry and a crooked tower. You can visit all of that in 2 hours (including lunch).
Florence, Tuscany and Umbria. Here it would be helpful to know if you are relying on trains or buses, or if you will have a car (which maybe you need for business reasons in Pisa, I don't know).
Until you chime in with that piece of info this is some info you might need.
Florence has severe traffic restrictions in the city center. Only residents with permit can enter. I don't advise to keep a car while in Florence, that would be a waste of dinero and a hassle. If you have a car, you either choose a location outside the city center, or even outside the city altogether, and visit Florence from there, or maybe you leave Florence to after the Tuscany's small town's visit (where a car is useful).
If you don't have a rental car, you should get one AFTER you visit Florence, to visit the Tuscan and Umbrian small towns and return it in Orvieto, before you go to Rome (by train from Orvieto, 70 min).
Tuscany and Umbria. You are changing hotels a lot of times with one nighters. Those are time wasters (pack/unpack/check in/check out), you should try to visit those places from a nice central base. For example if you choose a location near Siena and you have a car, you can visit Siena, Cortona, San Gimignano, Monteriggioni, Montepulciano, Volterra from that location. From Siena to those aforementioned towns, it's less than 1 hr drive.
Orvieto can be done as a day trip from Rome. But if you want to stay there, devote one day to Orvieto and one to Civita (spend a couple of nights in Orvieto). Assisi is also an easy day trip from Orvieto by car if you have the time.
Orvieto is where you should return your car, then take the train to Rome for your last 5 nights. If you don't have 2 nights for Orvieto, you could borrow one from Rome. I think 4 nights in Rome (=3 full days) are adequate your first 15 nights in Italy.
If you DONT have a car and don't intend to rent one at all, then you should plant yourself in Florence and visit all those towns (except Orvieto/Civita and Assisi) from Florence (via train and/or bus). Then you move to Orvieto for a couple of nights (via train) and visit Orvieto and Civita (and maybe Assisi if you have time).
Assisi is a little over 2hr by train from either Florence or Orvieto, so you could visit it from either in approx. the same time (Orvieto is a little closer though).
Then after Orvieto, on to Rome (by train)
