September is beautiful! The weather should be cool enough that you won't be dying from the heat, but the nights will get chilly, so make sure you're prepared for that. One of the things I also remember loving the most about the fall was the farmers markets. You could buy not only the freshest, most flavorful, veggies, but a lot of the vendors there would also be selling the jarred preserves and spreads that they had been canning. Delicious! Try to find some zacuscă - it's basically a vegetable spread made with eggplant, beans, tomatoes, and... some other stuff.
It's one of those things that every family makes differently.
I never drove there, so I can't really speak from personal experience, but at the same time, there was definitely a reason I never drove there! A lot of the roads are still not in very great condition, and if you don't know where you're going, it can be very easy to get lost. Taxis are plentiful though, and at least back then, were very cheap (and honest - in an entire year, I only ever had one problem with a driver trying to charge me more than the metered or agreed rate, and that was at a seaside resort where there were more tourists). When my parents came over to visit me, we hired an English-speaking taxi driver for the day to drive us from Targu Mures to Sighisoara, as well as to a local salt mine and to another village. For the whole day it was about $40. Granted, that was a little over five years ago, so I imagine prices have gone up somewhat, but they're still not going to be anywhere near what they are in western Europe or the US.
As for time, I would say to go for as long as you can! Ten days would, I think, be the minimum to visit a few cities and maybe get out into the countryside as well. (Oh! I just remembered - the painted monasteries in Bucovina are also super cool.) If you have longer though, there are so many interesting places to visit and it's still affordable enough that you can extend your trip without having to budget like crazy.
Good luck with the planning, and let me know if there's anything else! Congratulations on your wedding!