10 nights in Italy in late October - Weather??

The end of October is statistically the rainiest time of the year in Italy. However it is not possible to forecast the weather with any degree of reliability more than one week in advance. I suggest that you check the weather forecast just before you leave.
The best sequence is Venice, Florence, Vernazza, Rome. Siena is very close to Florence (45 min by car, 75 min by bus, 90 min by train) so I would visit it on a day trip from Florence rather than changing hotels just for one night. Checking in and out, with the packing and unpacking that goes with it, is time consuming. It's faster to get there and back for the day.
You should also rethink your plan to spend two nights in Vernazza that time of the year. If the weather is bad, you will be wasting your time. I would spend those nights in Florence and if the weather is decent I'd visit it for the day by train (2.5 each way from Florence). You already are not spending enough time in the cities you want to visit.

Even restaurants which are not strictly vegetarian will have vegetarian options in their menu. Italian cuisine is not traditionally heavy in meat and there are thousands of vegetarian dishes in the Italian culinary tradition.
Florence has a famous Vegetarian restaurant that I used to go to a lot in youth. It's called Ristorante Vegetariano and it is on Via Delle Ruote. It's an institution.

Yogi,
I was in the CT last October 5th thru the 12th. We would only have been "rained out" of the CT trails on one day during that time period. The day it rained we went up to Portofino and were able to hike the trail from Santa Margherita L out to the Portofino harbor and take the ferry back. That was the first day of our visit and the rest of the week was beautiful with rain predicted but did not show up and wash out our hiking the CT. For me if it were raining I would not have taken on the trails we hiked in the CT, they are rocky, muddy, with many up and down stone walkways and stairs so they would have been pretty slick. I have really good hiking boots that grip but would have had a ruff time on the trails we hiked. We hiked to Manarola to Corniglia, and Vernazza to Monterosso. Hiking boots were a big plus but as I said for me if it had been raining it would have been a real challenge to take on those trails. I've hiked in Scotland in the rain but in the CT the trails are different. But we lucked out on the weather as it was awesome, shorts most of the time while hiking. We also were able to ride the ferry up and down the coast, often rough seas can prevent tht. The ferry to Ponteverde was really a must do for me. We stayed out of the CI a bit up in Moneglia...we went down to Lucca (south of Pisa) on the 12th thru the 15th and it rained all three days off and on!

I have visited northern Italy in the mid-October through early November time frame and found the temperatures pleasantly on the cool side but with occasional light rain. Mind you, it was nothing to spoil a vacation but one must plan accordingly. Layering clothing and having a waterproof light jacket is the best way to go. I would also invest in good travel umbrellas but if you care to leave home without one, they can be purchased in ANY piazza in Italy as soon as the first drop of rain falls from the sky.

Note about gondola rides in Venice. During the autumn, a sweater or jacket is advisable. Most of the gondola ride will be in narrow "side" canals where very little sunshine actually reaches except at mid-day. And, even if you choose a gondolier somewhere around Pizza San Marco, you probably won't be in the Grand Canal long enough to enjoy a sunset (they duck into the side canals pretty fast to get out of the way, and wake, of the water buses). In all seasons except summer, this situation is exasperated by the fact that the sun is at a lower angle across the horizon. And, even if you choose a gondolier somewhere around Pizza San Marco, you probably won't be in the Grand Canal long enough to enjoy a sunset (they duck into the side canals pretty fast to get out of the way, and wake, of the water buses). I really think the pictures of couples riding off into the sunset in a gondola are mostly an advertising gimmick. But, don't worry, it's still a very pleasant experience.

That being said, I would also advise the north to south itinerary as well.

Thanks for the gondola advice and the advise regarding vegetarian food..I have to risk the 2 nights in cinque terre so will do it in the earlier part of the trip, as my understanding is the rains are more as October progresses..if it rains we will sit in our rented vernazza apartment and enjoy the local wine.. We travel real light and I want to see Siena after dark and before the day trippers arrive hence the over night plans..we are not so much into art so I suppose 2 nights in Florence should suffice..will surely see Michaelangelo's David and the uffizi though..will do reservation for the same as per advice on this site as our time is limited..