Hi there! I had a super long reply to all your questions and just lost it - so here a I go again ;-)
I think it is a great idea to divide the time between Florence with no car, then rent a car and stay in Tuscany. Whether in Montepulciano and that area really depends on what you want to see during the week. I suggest making a list of the places you definitely want to go see -- and then see if you can do them all from Montepulciano.... otherwise you could should to stay in the countryside a bit closer to Siena, to be a bit more central, if for example you want to make day trip to Pisa and Lucca, which are not that close to Montepulciano.
Hotel vs apartment in Florence:
I don't believe it makes much difference in terms of cost, it makes a difference mostly in space and in meals. If you get an apartment, you generally have more space... and if you have a kitchen, you can still decide to never use it and eat out at every meal.
The main difference would be in breakfast: what's your preference? Here, most Italians grab a coffee/cappuccino and a pastry for breakfast, so very light in terms of quantity... then they have a mid-morning coffee break.
If you love bigger breakfasts, then a hotel might be a good option as they often serve buffet style with both savory and sweet options.
Since you're planning on an apartment for the rest of the week, I suppose you could do hotel in Florence but it's up to you!
You can check apartments on our partner site
FlorenceAccommodation or on
Airbnb (this second one is an affiliate link).
Let me know if you have any more questions! I'd personally go for early October as a period