ITA flies from LAX and SFO (starting soon), allows pets in the cabin in business class, has reasonable charges ($240) for travel in the hold for the dog and no hard temperature polices, meaning you'll be able to get the dog on even in the summer.
They are generally easy to use with animals.
However, I would not make long-term plans involving travel -especially plans revolving around long-term investment in their FFP with regards to banking miles - on ITA as that airline is currently being lined up to be bought by LH. When that happens, pet travel policies will align with all other LH group airlines very quickly, the FFP will be absorbed into M&M and it remains to be seen if and how the US routes remain.
As there's very good high-speed rail networks in Italy, getting to Turin from Rome or MXP is easy and quick.and the preferred way to go. The trains allow pets of all sizes,

The same can be said for the second option - travel to ZRH with Swiss. There's almost hourly direct trains between Zurich and Milan , the journey takes three-and-a-half hours. You'd need to change trains in Milan, and from there, it's an hour on a high-speed service to Turin.Done that very often myself, works very well.


If you want to take the dog in the cabin on the transatlantic portion of the flight, there's only LaCompagnie to Paris, and trains from there.
However, there's no way your dog would be able to fly to EWR in the cabin with any airline domestically. You would need to use Alaska from the West Coast to Newark for the dog (Alaska is the only US domestic airline still accepting animals as checked baggage). That means separate tickets with no protection. La Compagnie is a very small airline with a great product for the price but less-than-stellar service recovery options. If things go pear-shaped you may get stuck.

Another possible option to fly your dog in the cabin could be on WestJet as "specialty dog". I have no personal experience with that but know people who successfully used them with non-service dogs. Not allowed in business class, though. They started flying to Rome and Paris from Calgary, but those are seasonal and not year-round.
They could be an option to get everyone to EWR in the cabin, connecting to LaCompagnie, though - they have service to PHX. IŽd definitely research this.