Taking you very seriously that what you want from this trip is relaxed meals, walking around, convenience, and minimizing travel time, and assuming that you picked Florence because you enjoy art, architecture, and history (even though you didn't say so), here's what I would do if I were you. This is based on my already having spent 6 weeks in Florence (and planning to spend another 5 or 6 weeks there) and also a week in Siena (and would happily return).
Fly non-stop Dulles to Rome and (if you can make this work) take the one train in the early afternoon that will take you from the airport to Florence without your having to change trains in Rome. Otherwise, train to Rome and then catch another train to Florence.
Stay for all 5 nights in Florence, somewhere a bit away from the very center. My husband and I like the Oltrarno and the Sant'Ambrogio areas. We greatly prefer renting apartments, partly because having only one small hotel room is the opposite of relaxing for us. if you choose an apartment with a helpful, English-speaking host, you do not need the help that people say a hotel will give you.
I so hate to say this, but just do a long day-trip by bus to Siena. Leave as early in the morning as you can and return to Florence after dinner. It is amazingly time-consuming and non-relaxing to move for one night. If you really really want to do it anyway, leave most of your stuff in your accommodation in Florence and just take what you need for a night and day in Siena, and stay in a hotel.
Others have mentioned some of the other wonderful towns you could visit (I would vote for Pistoia), but Florence is so packed with things to see and do that I'm not sure that more train or bus travel is what you want to do. If you decide otherwise, once you are there, you can do it spontaneously, without booking a tour or ticket in advance.
I would skip the wine tour. My husband knows wine and I certainly enjoy it, so we have visited wineries and vineyards and tried the local wines all over Italy, and I have to say that (here comes some heresy!) in general Tuscan wines are our least favorite. Plus, you don't sound as though a wine tour is your top choice. You can try some Tuscan wine in any restaurant or enoteca in Florence.
You really must spend your last night before your flight home in Rome. You can see some more of Rome, if you want to, or just arrive at the hotel in the evening. The Hotel Berna is a very close walking distance to the train station.
As for the time of year, both Florence and Siena will be plenty busy with tourists in May/June and in September/October --- there really is no "shoulder season" in those two places any more. But you can strategize. Visit popular sites very early or very late in the day, eat your main meal at lunch and take a long time doing it, and seek out the less famous but still incredible sites in Florence. If I had to choose a week, I would choose one in late October. I grew up in Washington DC, and I would do almost anything to avoid heat and humidity and tourist crowds.
