Most likely
No, that box transformed into castle
no. and that rainbow kingdom is Cloudsdale.
Yes, the Twilight's kingdom from My Little Pony Friendship is Magic is the Rainbow Kingdom.
The Crystal Empire is the Rainbow Kingdom in your Little Pony Friendship is Magic.
The My Little Pony Friendship is Magic Rainbow Power toys are connected to the box on the Root of the Tree of Harmony show.
Yes, the My Little Pony Friendship is magic episode Twilight's kingdom is involved in the rainbow power.
This information has never been revealed, so I can only guess. I would suggested watching the Season 4 Finale when it is released to find out. In my opinion, the Rainbow Kingdom and the Tree of Harmony box are related.
Yes, the official synopsis of the Season 4 Finale hints at the "Rainbow Kingdom".
The Crystal Empire is not yet considered to be the Rainbow Kingdom, but is suspected that is will be designated this title in the Season 4 Finale.
It has been suggested by the synopsis of the Season 4 finale and by press releases from Hasbro that the Crystal Empire is in fact the "Rainbow Kingdom". What this means is unknown.
It is not known what these flashes of rainbow in the pony's eyes is supposed to represent, but this will likely be revealed in the Season Finale of Season 4 of My Little Pony: Friendship is Magic.
This information has not been revealed yet. In the Season 4 Finale, this information will be available.