There's too much there to respond to it all.

It strikes me that there are quite a few sites on the Chuo / Sobu line that would interest you, so arranging to stay somewhere along that line might make more sense than your current digs.

These sites include
Nakano Broadway - Otaku stores / lots of manga memorabilia
Studio Ghibli
Akihabara
Tokyo Dome City (on a cos-play day)
I'd throw in the Edo-Tokyo Open Air Architectural Museum too - That might not sound like it would be thrilling, but show the kids the website. It is actually very cool.

Since you're doing this on a budget, I'd be inclined to skip the daytrip to Kyoto. Take a day trip to Kamakura and perhaps Enoshima. If you want the kids to see Fuji, make a trip up to Kawaguchiko. Enjoy the rollercoasters at Fuji Q Highland. There's a high risk that Fuji will be entirely obscured by cloud on any given day in June, but ...

All of this makes me feel that Shinjuku would be the location to stay. But YMMV.

Disney is the "odd one out" in terms of location. I'd plan on doing it last to cap off your trip. Take the bus from there to the airport.