I would work your itinerary in reverse. Fly into NAP, rent a car, and head out to Pomigliano then up to Anzio via the A1. Drive north towards Rome's GRA and take that east, exiting towards A24. That will take you northeast across the country through L'Aquila all the way to the east coast. Head north on E55/A14 for a scenic drive and that will take you into Ancona and the Jesi area. Continue north to Misano Adriatico. Since you're finishing here than that would leave Cremona out, since it's closer to Milan than where you'd be. Fly from Rimini's Federico Fellini to FCO for your return flight.

You're other option, if you really want to go to Cremona, is to fly into MXP and drive southeast, with Cremona being your first stop. You can work your way past Misano down to Ancona & Jesi, then come back up to Misano Adriatico and head out the same way from FCO via Rimini. This leaves out Pomigliano & Anzio.

If you're determined to see them all then I salute your persistence. In that case I would pick up your dog in Misano and head north, stop at Cremona, and out via MXP.

Buon viaggio!!