Hi again,
Gala Yuzawa is open until the start of May, so you can certainly visit in April. Takasaki is on the way back to Tokyo, so you can always transfer there to a E5 Shinkansen, in case you would like that experience too.
The best way is to simply buy a ticket at the station in Japan, you can tell the staff what you would like to do (for instance try a specific train type, as what kind of seat you like (upper/lower on E4) and select a time slot. This way you will be sure of getting the correct ticket. There are also departure boards around the station that display the train time (eg. Toki for E2 Shinkansen and Max Toki for E4 Shinkansen).
There are different transport systems in Tokyo, there are the local JR lines (which you could see as Overground) and there is the Metro, there are buses, Monorails, private lines and more. You can buy a JR ticket for the whole of Tokyo or a day pass for the metro. These generally offer good value if you plan on exploring a lot (more than 4 rides a day). The JR Pass also covers all JR lines in Tokyo. Disneyland can be reached by JR line from Tokyo, the station name is Maihama.
Hope this helps,
Daniel