I am a slow traveler so for me this itinerary is too fast. I am sure some faster travelers will comment as well, so you can get their input as well on the speed.
10 landing in Tokyo at 12.55
11-12 Tokyo
13 to Koyasan, visit and sleep there (Candle Festival) did you check how long it takes to get from Tokyo to Koyasan? It already takes quite long from Osaka. You need some time on location. Koyasan is a nice place. Which deserves at least 24 h of staying there.
I donft know about the candle festival, Ifd assume it is in the evening, but as it gets dark early in Japan eben in sind it might not be as late in the day as you think.
14 to Nara, visit and sleep there (Lantern Festival) Nara and Kyoto are very close. I would stay in Kyoto that night. Theoretically you should be able to leave your luggage in a locker at the station , however with a festival locked places might be harder to get. What I would do, send your luggage from Tokyo to Kyoto and only bring one small luggage for the entire family (or everyone a small backpack) for yes one night at Koyasan. Luggage shopping within Japan is very common.
Further on luggage, pack as small as possible. Hotel rooms are small and also in trains there isnfta lot of space for big luggage. Best is carbine size luggage. HOWEVER if you are going for the shipping option, packing 2 big suitcases for the entire family might also be an option. Shipment prices are per piece (and size).
Many hotels have washing machines (and dryers) so it is enough to have clothes for about 5 days.
15-18 four days in Kyoto
19 to Hiroshima, visit (including Miyajima) and sleep there
20 day trip biking the Shimanami Kaido, back and sleep in Hiroshima (feasible ? long day for sure..) it depends how much of the Shimanami Kaido you want to bike and obviously your speed. I did it a few years ago and it is a wonderful place that deserves time. I took 2 days with a nice stop in between and sightseeing en route.
http://bicycletraveljapan.blogspot.com/2017/11/part-10-day-48-49-ridin...You will need to reserve the bicycles. Also there are generally 2 types of bicycles for rent: racing bikes and gmamacharih . A mamachari type of bike can be fun but canNOT be fast. And they are quite uncomfortable. Not a problem if you only intend to cycle a small part, but a problem if you want to make km. And frankly the nicest part is in the middle.
As you have some extra days, I would add a day here. Potentially I would also do it before Hiroshima as it is slightly backtracking.
21-23 to be defined (see below) I would stay in the region where you are already. Eg Naoshima, Teshima (there are some interesting museums in the countryside), Inushima. Or over to Takamatsu on Shikoku and see the splendid garden.
24 sometime during the day, back to Tokyo (visit ?) and sleep there
25 Tokyo
26 Tokyo + take off at 17.45Being from Italy I am not going to want you of the humid heat. But in my experience Japan feel hotter in summer than Italy or Spain (I lived in both countries for several years), because of the humidity.
Enjoy planning your trip.