Of course, everyone has different experiences with weather. I did the Village Italy tour last fall starting around Sept 29 and ending mid-Oct. I thought it was pretty hot still, particularly in Venice. It was rainy at the end of the tour and the waterproof jacket came in handy, but we ended near Milan instead of Rome. Still, if I were going ~any~ tour in Italy (or Europe for that matter) I would want a waterproof layer. My Marmot Precip with a cardigan would be enough warmth for me.
I did leave a winter coat in my car in case it was cold when I got back to Idaho as I've got about an hour's drive home.
I am glad you are not worried about solo travel on a RS tour. I've done 5 tours, 2 of them solo and had a wonderful time!
I don' t think anyone has mentioned it yet, but do be aware in Venice the bridges you see are actually steps not ramps so you will have to carry your bag up and over any that are between the vaporetto stop and your hotel. Also, if you are using the train for your extra days, hefting a bag up the steep steps into a train is a chore as well. Just throwing this out there for you to consider so you can get your bag weight as low as possible.
Buy your shoes soon and start walking in them. In Rome, particularly, be ready for going 5+ miles in them. I did the 21 day Best of Europe and we had huge walking days in Rome and Amsterdam....10-12 miles each day! You will be in Rome longer so may not have that much walking in a single day, but do make sure you can go at least 5 miles in them. It feels like the Vatican Museums are at least 5 miles thru to the Sistine Chapel!