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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.
Yes, the My Little Pony Friendship is magic episode Twilight's kingdom is involved in the rainbow power.
The My Little Pony Friendship is Magic Rainbow Power toys are connected to the box on the Root of the Tree of Harmony show.
Most likely
No, that box transformed into castle
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.
Probably not. It has been said by the production team of Equestria Girls that Equestria Girls was meant to be a standalone, and not be connected to the TV show My Little Pony: Friendship is Magic.
This information has not been revealed yet. In the Season 4 Finale, this information will be available.