Hey Arjun, let me first tell you how jealous I am......I have always wanted to go to Italy, but no luck so far.
From what I have heard from my Italian friends though, Florence is quite expensive as far as cost of living is concerned, so maybe you can post a query in the apartment rental section on this forum. The site itself (
www.lifeinitaly.com) has also opened a new section for rentals.
I have travelled to other places, and I find vegetarian food is easy to find, and Italian cuisine does have its vegetarian fare as well. But if you are a true jain and do not eat even onion and garlic, be ready for some very difficult times......
I do not know how the customs will take it, but carrying stuff like besan ke laddoo would probably be a good idea.....but I dont think you can pack snacks for the duration of the stay because you are going in October and if you stay the winter, you will need to pack in a lot of winter clothing.
I think an Italian could give you good tips on where to find vegetarian food, and also an affordable place to stay, but had to put in my two cents

. I am sure some friendly Florentine on this forum would come forward!
But if you are in Florence, and are somewhat of an art-buff, you could not be in a better place.....and even if you dont love art, you are bound to fall in love with the art in this city.
Buon viaggio e in bocca al lupo, (Wishing you a good trip and good luck)