In truth, nobody can really answer your first question - do you want to pick a few of the best sights in a lot of places, or thoroughly see what an area has to offer?
If you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5 PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are wonderful.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jkgQdiQlK2bcKCzr2pJQh2
Kyoto and Nara are the crown jewels of traditional Japanese culture, and well worth a lot of your attention. You could reap maximum savings by going down to Hiroshima or further and back within a week on a 7 day JR Pass. Or if you only want to go down to Osaka/Kyoto/Nara, then a 7 day Japanican discount ticket would be far cheaper for you.
http://www.japanican.com/en/tour/detail/VJOPENTO1/
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jHC27aeRwEHaP7Pw_G277w
Nara is one place you should not skip either - the Todaiji Temple in one of the finest in Japan.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jGyApXWSTIKwkioxuAKwt5
Himeji and Hikone Castles are both excellent and originals - not fake reconstructions. An overnight at Koyasan would be an unforgettable spiritual experience for you too.
I suggest you spend a week in the Tokyo area (Kanto) and after 3-4 days of seeing Tokyo, go on some day trips to some very good nearby places like Nikko, Kamamura/Enoshima, Hakone, or Karuizawa. A Tokyo Wide Pass can save you a ton of money over its 3 day validity. A Tokyo 3 day subway pass can save you some money too.
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03
The rest of your time can be used for the Kansai area and Hiroshima if you like.