No no no, you don't want to do Kyoto from Tokyo! Kyoto is over 2 hours from Tokyo on the Shinkansen Nozomi, but only 30 minutes on the JR from Osaka station! It doesn't make sense to do Kyoto from Tokyo, both time and cost wise!
If you want to do Kyoto, I would recommend cutting short one day in Tokyo and use it to do a day trip to Kyoto or Nara from Osaka instead. Between Kyoto and Nara, I think Nara is more doable as a day trip than Kyoto, because there is way too much to see in Kyoto, and doing Kyoto in 1 day is in my opinion, not doing this beautiful city any justice. With your young kids, Nara is more manageable with the 3 main temples and they will probably like the deers that roam throughout Nara Park. (The deers can be aggressive for food if you choose to buy the deer biscuits from the vendors there, so be careful).
Regarding your last question, the Shinkansen departs from the Shin-Osaka station, not Osaka station, so u will need to take the local JR to Shin Osaka station to transfer to the Shinkansen. You're right re transfer from Tokyo station to Maihama. From Maihama to Shinjuku, there are a few options -
(1) You can go back to Tokyo Station and take the Chuo Line (JR) to Shinjuku station or the Marunochi Line (Subway) to Nishi-Shinjuku station (Slightly nearer than Shinjuku station to the Hyatt)
(2) Take the bus from Tokyo Disney Resort to Shinjuku Station. This option is more direct, but it drops you further at Yoyogi station (far south of Shinjuku station), and to walk from the drop off point to the Hyatt Regency would be quite a trek.
http://www.tokyodisneyresort.jp/en/access/bus/shinjuku/