@LikeBike, the only new place I threw out was Hiraizumi and that was because OP stated they wanted sakura originally, but only as a possible option. Every place else I mentioned was someplace OP already wanted to go. Perhaps you are responding to someone else?
OP, there really are no 16 or 10 or 25 Must Dos, it's all about what you think your group is going to enjoy and finding a way to enjoy Japan. It's OK to go to Japan and just do amusement parks. People do. None of them are the same as US ones. I'm from NY and yes, I've gone to USJ and Tokyo Disney. USJ is cheaper for me than going to the Universal in FL or CA. My entrance ticket for me and my daughter is the same price as my entrance ticket for just me in the US and there are rides in all 3 parks that are unique to Japan. But I also don't go to Japan with only 7 full days with a kid or a long list of things I want to do. My daughter's shortest trip was 10 days, and the last two trips have been over 2 weeks. So it's easier to fit in an amusement park as well as other touring ideas and I always try to balance what we do since I know we will be back. There is also plenty I've done in Japan without my daughter since I went from 1999-2010 on a regular basis. So this trip she will finally go to Fushimi Inari, but I've already gone 2x.
This is the crowd calendar for Disney Sea: http://www15.plala.or.jp/gcap/disney/
You want to go to Sea on either the 22nd or 23rd. Sea gets crazy crowded because it's both a family park and a date park. You would need to pay me to ever go on a week end again.
A hint for traveling with kids, get them their own cameras and encourage them to take their own pictures. My first trip with my daughter (when she was 5) she got super whiny at shrines and temples because she got bored quickly. Now she loves visiting them with me (as well as gardens) because she has her own camera and gets to take her own pictures.
You could try renting bikes for getting around Kyoto. It's not going to be super relaxing since you'd be biking in a city, but it's a way to try to fit biking in. There are some wonderful places to bike, but again, you only have 7 days which really limits what you can accomplish.
My daughter and I didn't enjoy the Ghibli Museum and my daughter loves Miyazaki films and I'm glad we don't need to go back.
No, there is no sakura in Tokyo that late in the season. You'll be getting azaleas and possibly wisteria. The only places to see sakura that late are several hours north of Tokyo and with 7 days you don't have the time unless you drop something like Disney Sea. I think Kurama in Kyoto has some late cherry blossoms but it's pretty far from central Kyoto so you would need to decide that is where you are going your Kyoto day. I went in late April about a decade ago and there was a special type of late blossom that grows there, it doesn't look like the traditional sakura though, look up Yaezakura or Kanzan to get some idea. Azaleas are quite pretty and so is wisteria.
I would stay in Kyoto vs. Osaka. If you book quickly I've seen more apartment hotels in Kyoto vs. Osaka and you are a large 5 person group. There are a bunch of really cute looking places meant for families there. I 'm just going by images on websites though. I'm a party of two (me and my daughter my husband stays home) so renting a large unit is overkill.
If you get up early you should be able to see Mt. Fuji looming over Tokyo. You also go past it on the shinkansen between Tokyo and Kyoto. Actually visiting Mt. Fuji is going to take a lot from your schedule and you don't have the time.
My daughter loved the cloud coffee shop in Kyoto. We also went to the official Pokemon Cafe in Tokyo. Whenever we are in Tokyo Station we buy the 400 yen donuts that are super cute. I keep meaning to take the 30-40 minute trip to buy Totoro cream puffs.