I've done your locations multiple times in the past couple years - last summer with a first-timer. I can give you my impressions as a non-foodie, non-shopper, most often solo traveler.
Looks fast, but OK overall although I'm not sure where your travel is allocated - at the end of the day range indicated, or the beginning of the next. My comments below are based on where the travel time might be placed.
Kyoto and Osaka are commuting distance. I would consider combining them and would suggest staying in Kyoto (lots of in town and short trip outside as special sightseeing spots) with a day trip to (or thru) Osaka if you have specific interests there.
That is the next question - is there anything in Osaka that particularly "calls" you? It is known for its street food and nightlife. I, personally, recognizing I am NOT you or many others, have found other sight seeing opportunities more interesting and compelling than Osaka. (On the other hand, I have had business trips there and enjoy the city aspect.) Osaka's Castle, for instance, is rebuilt. Although its museum is fine, Himeji, that you hope to visit, is an original. Dontonburi is interesting enough, but food and shopping related. (My first timer loved it last summer - I know my opinion of I-can-take-it-or-leave-it is perhaps not the norm.) The main reason I suggest examining Osaka for your specific interests is the length of your trip - perhaps the time could be allocated back to Kyoto - or Hiroshima.
Himeji can be visited from Kyoto with just a little more travel time investment than from Osaka. I would allocate minimum of a full morning, starting out early, but you can be back in Kyoto by the early PM, or on the return stop in Osaka for an exploration.
Hiroshima- not quite sure of the length of your stay. One day to see Hiroshima's Peace Garden, Memorial, and Miyajima, not to mention other interesting spots is not enough. Maybe you were planning a half day on one of the travel days - one day for Miyajima and one half day for Memorial?
Train tickets - there should be no problem buying the day of or a day in advance in convenient. No difference in price so no advantage of early purchase. Lines, if you want a live person, can be quite long. The ticket machines have English options and are not difficult to use - particularly if you have studied the train options and know what you want. Check the Navitime and Jorudan apps. Easy search options. Train tickets can be purchased with credit card from a live person, machines may not take a card. I always carry some cash so I can purchase from a machine, but that is my personal quirk.