As my username might give away, I tend to use a bicycle also for traveling in Japan.
Definitely the region around Asuka, Sakurai etc is convenient with a car (or a bicycle). I used small streets but also on the big ones there isnft that much traffic. I went to Hasedera in early November 2 years ago (by bicycle) and remember a large parking lot. I remember it because they made me pay even for my minuscule brompton, which I thought was a bit of a theft, but wasnft willing to argue.
http://bicycletraveljapan.blogspot.com/2017/11/day-42-roundtrip-throug...http://bicycletraveljapan.blogspot.com/search/label/NaraFor driving in Nara, as far as I remember from a bicycle saddle it didnft look so bad either. I wouldnft GO by car to sightsee in the center but if itfs only a mode of access it should be fine. If you have a car I would however consider to visit the more fared flung places of Nara instead. Nara city you can always visit w/o a car, Horyuji area is more complicated by public transport and really beautiful:
http://bicycletraveljapan.blogspot.com/2017/11/day-41-nara-to-yamatoya...For cycling around Kyoto, maybe at some of the most famous temples it can be hard to find a parking lot for your bicycle but when I go nowadays I go to smaller locations and never had a problem parking my bicycle.
Here some locations in Kyoto I visited by bicycle:
http://bicycletraveljapan.blogspot.com/search/label/Kyoto?m=1For train from Nara to Kyoto both jr and Kintetsu are convenient. Google maps will tell you which is easier for you to reach from your hotel. Kintetsu is a bit more central in Nara. I never reserved a seat. I think itfs mainly local commuter trains.
If you want to do a Shinkansen trip someplace like Himeiji, Hiroshima, Okayama ... would come to mind. Many people do reserve seats on Shinkansen, I personally generally donft (specially between Tokyo and Shin Osaka) because I travel alone and prefer the flexibility to just turn up at the station, buy a ticket and get onto the next Shinkansen 5 minutes or less later. No need to rush. No need to wait. If I still want to buy a bento, I can simply buy it and take the next train. I personally find that convenient but all my colleagues think I am crazy. Company would actually pay the reservation fee and even the green car fee ( which is reserved seats only).
However even if you do get a reservation if you are at the station way too early you can easily change it or just hop on unreserved car of an earlier train (you should really change it, so JR can make the seat available again) and if you are too late you can also just take an unreserved car. Note that there are certain Shinkansen which are reservation only, but none of them in the Kansai area as far as I am aware.
Enjoy your trip to Kansai!
Enjoy your trip to Kansai!