Amalfi, Puglia and Sicily in 2 weeks?

Might want to change your title, 3 days each. It will be rushed, but doable. I think you'll be fine for Amalfi

Take a look at Ortygia/Siracusa for Sicily.

Where are the rest of the two weeks? If you could add them to this, it would be a lot better, but this is "doable" if you mean four nights for three full days on the AC. I would strongly suggest adding the Sicily time to the AC and Puglia--see Greek temples at Paestum, and see Pompeii's theatre instead of Taormina for example.
Monopoli is a nice base for Puglia, but any of the Itria Valley towns work.

It seems too rushed to me. For one thing you lose so much of a day when you fly. We just had 9 nights in Sicily, then flew back from Palermo to Rome. Flight was delayed and a 1 hour flight literally took up 6 hours of travel time by the time we checked out, security, waited etc. Personally I would either drop Sicily or prioritize it. You could combine Naples, Amalfi and Puglia and eliminate two days of flying.

I recommend getting a calendar and researching travel options with rome2rio.com. That might give you a sense of where you need to go without really rushing it. I am doing Southern Italy for the month of November, have been to Sicily and know what it is to want to see it all.

I would recommend either staying on the west coast and Sicily and dropping Puglia or doing mainland Italy since you are all keen to go to Alberobello and drop Sicily. From my research Aberlobello is intriguing and special but extremely touristy and recommendations say 3 hrs max for a visit.

I'm only counting 9 days or nights? in your itinerary. That's not two weeks, more like a week and a half.

Regardless, this itinerary is in my opinion too fast-paced and, more critically, too far-flung to work. Any missed connection, whether train or flight, or the three hours most rental agencies are closed in the afternoons, and so on, will seriously cut into your sightseeing and relaxing time. Have you researched flights and factored transportation time into your plan?

I have been to all these areas. If it were my trip, I would either cut Sicily or cut Puglia or AC.

Also, driving on the Amalfi Coast is a bit of a drag and September is not off-season, so you will be sitting in lots of traffic. Alberobello is quite touristy and there are trulli dotting the countryside throughout the Valle d'Itria, so while you may want to pop by Alberobello, you don't need to make it long visit.

It’s absolutely too much. By super-serving everyone’s separate travel dream, you’re really setting yourselves up for a rushed sightseeing trip that includes a lot of travel time between places. That time spent traveling is time lost, and you already recognize you don’t have much to start with. Even though it’s a short flight from Naples to Catania, you have picture that day from start to finish: check-out, transport to airport, eating/flight, transport from Catania to Taormina, check-in… it’s a full day and you haven’t done a single thing. Ideally, you simplify this down to 3-4 destinations (accommodations), giving you 3-4 nights in each place. Plenty of time to explore, relax, day-trip… experience the place.

Luckily, your primary target of Amalfi Coast region has enough to satisfy everyone and is as an extraordinary destination as you could ask for. I recommend staying for part of the trip in either Naples or Sorrento, which make excellent bases for exploring sights on the Bay of Naples (Naples/Pompeii/Herculaneum/Capri/Ischia.) Then switch to a town on the Amalfi Coast itself. By staying in a coast town you really crack the code on crowd avoidance and those days become really pleasant. You’re basically always moving in the opposite direction of the day-trip crowds, which are significant. Magnificent vistas a-la-Taormina? Check, and in spades. Beach? Check. Small town ambience? Believe it or not, in its off hours the Amalfi Coast can feel very local. In upper Positano, the residents all greeted me with a “Buongiorno!” and in the small town of Atrani folks hung out on the square to enjoy a gelato while their kids played. Your travel dreams are maybe easier to get hold of than you think.