Background:
I've been invited to join family on a trip to Japan this September (mid/end Sept - mid Oct [NB: I know this isn't the best season but the dates are firm]). We'll have 17 full days in Japan not including travel there/back and will be based in Tokyo, Osaka and Kyoto. There'll be 2 kids on the trip aged 9 and 13 so aiming to cover a mix of fun and amusements and cultural immersion.

Recommendations wanted!
There are already a list of things planned for the trip that the travel agent has put together.

For those that have been, I'm wondering what are the things not to miss in/near the cities we'll be in that aren't already in the list, or the experiences people had that they'd highly recommend we do?

Things already planned:

• Cultural: Sumo Stables visit, Geisha meet and greet in Gion, Tuna Auction (if we get tickets), Ninja Training experience
• Museums: Manga Museum, Cup Noodle Museum
• Wildlife + Outdoors: Nara Deer Park, Nagano Hot Springs, maybe Okinawa Aquarium (with a few days at a resort there)?
• Theme Parks: Disney Sea and Universal Studios

Thanks in advance!

You might look at Gion Corner in Kyoto.

Have you consulted japan-guide.com? it's an excellent resource.

When I went with my kids, 13 and 16, we did go to Disney Sea which is indeed very awesome. Make sure you get there for rope drop. DIsney Sea is unique unto the world, not so much the case with Disneyland. I did not go to Universal.

We also went to see 3 sites in Yokohama: Sea Paradise (a big aquarium complex with a few spectacular amusement rides), TaYa Cave (not a show cave, carved out by monks and they give you a paddle with a candle on it. You can 'recharge' the candle inside the cave where they have more candles). And the Yokohama Ramen Museum which is a two-story recreation of a Tokyo street scene from the Showa period. It is basically a great excuse to eat ramen as they have at least 7 famous ramen vendors from throughout Japan. You can get half bowls of ramen so you can sample a variety. All three are spectacular places.

In Tokyo we went to Sunshine City Ikebukuro, a multi-skyscraper complex which houses a shopping mall, an indoor aquarium, a planetarium, a 60th floor observation deck, and most uniquely an indoor theme park called Namjatown. Everything will be in Japanese, but ti's still fun and there are two food ares of special interest: Gyoza Stadium with at least five gyoza venders serving many different styles of gyoza (ask for the shredded garlic gyoza) and an area with fanciful desserts.

>> Sumo Stables visit

You could go to the actual September sumo tournament in Tokyo:
Grand Tournament Schedule - Nihon Sumo Kyokai Official Grand Sumo Home Page
That would be noon or 1PM until 6PM. I can give more details if interested.

>> Nagano Hot Springs
Makes me think of the snow monkeys in a Nagano hot spring.

>> maybe Okinawa
Maybe. Check the weather.

You've listed pretty touristy attractions/activities. That's not necessarily bad, but instead of adding more "do not miss" attractions - which usually turn out to be super popular and touristy - I'd make sure to leave time for wandering and serendipity.

In Tokyo you may want to walk to one of the beautiful parks/gardens and let your kids get drinks from the ubiquitous vending machines, snacks from conbini, and toys from gachapon machines along the way.
In Kyoto wander a neighborhood and go to a couple of the less well known but still beautiful temples. Your kids might enjoy the Okazaki "rabbit" temple. I believe there's another one dedicated to mice or rats.
In Arashiyama (western part of Kyoto), you can go to Otagi Nenbutsu-ji temple, which contains hundreds of unique (and fun) rakan statues. The walk from the main streets of Arashiyama up to the temple is through a nice neighborhood with some beautiful homes.
It might be fun to notice what Japanese kids are doing for fun and spend an afternoon doing something similar - hang out in an arcade or gachapon shop, wander through a mall, or go to a karaoke place.

Since there'll be 2 kids in tow, I expect we won't have too full of an itinerary and there'll be a fair bit of time to meander around. For that same reason, I suppose it'll be more 'pop culture' than 'culture' we take in so that it can be fun for the kids (I'm hoping to balance it out with a few more sites of cultural importance!).

The historic district of Asakusa is an area of Tokyo that is both fun and culturally rewarding. Best of both worlds. It can also include the aforementioned meandering.

You're hitting the September sumo tournament in Tokyo (Sept 13-27) which is actually perfect timing. If you can get tickets, I'd go for that over the stable visit. The stable tours are interesting, but tournament day at Kokugikan is a whole different thing - it's loud, it's long (afternoon session runs 2:30-6pm), and nine and thirteen-year-olds usually get pretty into it. You can probably squeeze in both if you have the time.

For Tokyo with kids - Akihabara's got the big Pokemon Center and they'll probably want an hour minimum just in there. Takeshita Street in Harajuku if they're into the fashion thing. Miyashita Park in Shibuya has skateboarding and climbing walls when everyone needs to burn off energy. Conveyor belt sushi (Kura Sushi or Sushiro) is good for kids this age - they pick their own plates and it keeps them occupied.

Early September can still be pretty hot, and you're at the tail end of typhoon season. You get several days warning when one's coming, so just pack a light rain jacket and you'll be fine. By October it's way more comfortable. September's when the autumn festival season starts, and honestly the random neighborhood matsuri that pop up are better than the big tourist ones.

Fushimi Inari and Arashiyama get stupid crowded midday. I mean hundreds of tour groups. Go at 7-8 AM or late afternoon. Kids tend to like Fushimi Inari more than they think they will - there's fortune drawing, collecting the little charms, climbing up through the gates, lots of fox statues.