Your plans are not unusual and no real red flags, although going from Hiroshima to Tokyo in one shot will take 4¾ hours with a better connection (presuming you are using a 14 day JR Pass).
One day for Hiroshima and one day for Miyajima should be fine - I don't see any point in taking a day off of Tokyo and add more. Tokyo has loads to see and do - did you look through the site's page on the city, as well as the official city tourist guide, etc?http://www.japan-guide.com/e/e2164.htmlhttp://www.gotokyo.org/en/index.htmlhttps://goo.gl/tAECfM
Seeing the Sensoji Temple and Meiji Shrine (and don't skip the Meiji Gyoen Garden there also) is pretty much de rigueur for any first timer. There are close to a dozen beautiful historic gardens in the city, some with a history going back centuries. Plus lots of places for great city views - the Sky Tree and Mori Bldg are some of the best but also rather pricey. If you want to save some money, there are also many good places that are free.
For your Day 2, you could see Nara Park, which has most of the best, in half a day, and then see Osaka if you like. For Day 3, you could see Himeji Castle and the nearby Kokoen Garden (get the combo ticket, and one is virtually free) also in half a day, and if you like, you could also go see Kobe for half a day as well.https://goo.gl/65ZtRi
For Kyoto though, you can avoid the huge crowds at Fushimi Inari if you go very early in the morning or evening. Actually same thing for the Bamboo Grove, though the shrine is worse. It's not hard to get from the shrine to Arashiyama by train via Kyoto Station. But just going there for the grove (you'll be through it in 20 minutes or so) and the monkey park is kind of a waste. There are many other sights in the area that alone might not be worth the trip, but all together collectively make a good outing. The Otagi Nenbutsuji Temple at the end of the line is one of the best and unforgettable.https://www.japan-guide.com/e/e3950_arashiyama_full.htmlhttps://goo.gl/Neiowd
For your free day in Kyoto, you are missing lots of the big ones - Kinkakuji, Kiyomizudera, Sanjusangendo, Nijo Castle, etc. Pick what interests you the most.http://www.japan-guide.com/e/e2158.htmlhttps://goo.gl/knQtMg
As for seeing a geisha (most likely a maiko), the place to be in Gion in the evening - but no promises. If you want something definite, there are performances you can attend.http://www.kyoto-gioncorner.com/global/en.html