Raincoat

Yes.

It's best to have a rain shell that is packable (not lined at all) then add layers underneath - light wool sweater, t-shirts, lightweight fleece - for warmth as needed.

Unless you get into higher altitudes, I would expect warm days and cool nights, as a rule, in Northern Italy in September. There may be unseasonably warm or cold weather so you want to pack with as much flexibility as possible.

Ditto. You'll want something very light (in fact, you cold even just buy one at one of the "euro" stores here, around Rome most of them sell decent ones for one euro. Then you could always just throw it out if you don't want to carry it home.