Not that it's a definitive answer, but I notice that the Best of Italy tours, which start in Varenna on Lake Como, have tours starting as late as October 16th this year. So some things may close, but I don't think it would be totally dead at that time or they would probably not be running the tour that late in the year.
That said, when we were on that tour at the beginning of October a couple years ago, they had just changed to the fall ferry schedule on October 2nd, so there's definitely a "change" of some sort at that time of year. What you might want to do is figure out what sights you want to see and check their schedules to see when they close for the season. Ditto with hotels and restaurants. Most places have websites, plus I know there's schedule information in the Rick Steves books and other travel guides which should at least give a general idea of whether they close by mid-October.
We did have one of our two days of rain on the tour when we were at Lake Como, but it wasn't heavy or steady. We ate lunch outside in Bellagio, and took a long walk on that day. Mostly it was waiting for the ferry and the ride back to Varenna that was "wet". We choose October for our tour because the weather seemed to be neither too hot nor too cold, and the average number of days without rain (in Milan anyway) was 23, so that was a 3 to 1 ratio of non-rainy weather. Another website for Lake Como itself shows an average of 12 rainy days in October, but as you said, no one can predict the weather. We were very lucky the year we visited as I expected more rain than we experienced.
