First time to Italy, will stay in Florence/Siena. Multi-Questions

I highly recommend Rick Steve's Italy guidebook. It helped me immensely for my first trip to Italy. It gives great time- and money-saving tips and comes with a handy map. He also breaks down the city into different areas with suggested hotels (I don't think he really mentions apartments but he does give out website info for apartments), so you can also see which sights are close in relation to your hotel or apartment. In Florence, I stayed in a B&B in Piazza dei Ciompi. I wouldn't say it's super central but still very walkable to the sights I wanted to go to. Then I carpooled with friends and took a short trip to Siena, where I wish I could have stayed longer. It depends on what you want to see in Florence and Siena, then I would select an apartment close to where you want to go. If you just plan to stay in Florence, it is walkable and there are buses. I remember Siena also being walkable but I wasn't there for very long. If you want to venture out to the surrounding towns, then I would rent a car. I don't know about the Florence airport, as I took a train from Rome. However, Rick does give good information on airport and other connections in his guidebook. Have a wonderful trip!

Wife and I were very happy with the following places--both very comfortable, quiet for sleeping, but still close (5-10 minutes' walk) from the "thick of it":

Florence:
https://www.airbnb.com/rooms/2702043?eluid=1&euid=2ba8364f-0b7f-ed8f-6ab8-e59e76ccab4c

Siena:
B&B La Coperta Ricamata
Via Garibaldi 46, Siena, 53100, Italy

I can't comment on your other questions, except to say that we took the bus from Siena to Florence rather than a train; it was a bit of an adventure finding the right bus at the large outdoor station (no more than 5 minutes' walk from the recommendation above BTW), but once we did, the trip itself was fine, and the drop-off location in Florence pretty convenient.

Start with Rick Steves' Guidebook for Florence and Tuscany. You may want to read others, too, but he covers the area so well, along with advice on transportation. More down-to-earth than many guidebooks.

How long are you planning to stay? What do you want to do: Tour wineries, experience Renaissance art, explore historical sites, hike?

1. That is what I do when I go to visit my family in Florence (once or twice a year). However occasionally I fly with mileage awards, and finding mileage award seats to FLR is not as easy as Rome or Milan, so in that case, I fly to Rome and take the train to Florence. But when I pay for the ticket, unless I need to go to Milan or Rome for other reasons, I always fly to Florence. The ticket to Rome/Milan would need to be very significantly cheaper to make it worthwhile for me. Trains aren't cheap and time has value too, especially because I would need to spend the last night in Rome too, since flights to the US are in the morning. FLR has no flights to North America, so you would need to connect to a European hub (FRA, MUC, AMS, CDG, ZRH, FCO). From IND I don't think there are direct flights to Europe, so if you fly to FLR you would need to change planes twice. In such case it might be preferable to fly to Rome or Milan (via New York or Chicago or Atlanta or other major US hub) and then take the train. The train from Rome FCO airport to Florence takes just over 2 hours (likely one change at Roma Tiburtina). The train from Milan MXP to Florence takes nearly 3 hours (one change at Milano Centrale).
2. The FLR airport is so close to the city of Florence that it is walking distance (actually it is inside the city already). But if you stay in the historical city center (which I would recommend), then a taxi is recommended (3.5 miles for about 25 euro), because it would be a long walk. There is also a shuttle bus from the airport to the station (6 euro per person). Either one takes about 15-20 min depending on traffic. Siena is about 1+ hr drive from the FLR airport. A taxi or private car would cost you about 130-140 euro. If you use public transportation from the city center of Florence you can take a bus (8 euro) or the train. The bus is preferable, because it's 15 min faster (75 vs 90 min) and drops you at the city center of Siena on top of the hill (the Siena train station is at the bottom of the hill).
3. Anywhere in the historical city center of Florence will fit your requirement. The core of the historical city center is only about one square mile, so very walkable and flat.
4. While staying in Florence, a car is not only unnecessary, but not recommended. The entire historical center is offlimits to cars (except for residents with permit and service/emergency vehicles). While in Siena, a car could be useful if you stay several days and plan to take day trips to the small towns and country side of the province of Siena. In that case, make sure your accommodations are outside the Siena city center, or even outside of Siena altogether. Siena's historical city center has the same car restrictions as Florence, that is you are not allowed to drive inside of it, unless you are a resident with permit.

If you plan to rent a car research about European driving signs and rules, ZTL (Limited Traffic Zones), International driving permit, etc. You can use the search feature above to find plenty of posts on that subject.