Yes, the Twilight's kingdom from My Little Pony Friendship is Magic is the Rainbow Kingdom.
no. and that rainbow kingdom is Cloudsdale.
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.
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.
This information has not been revealed yet. In the Season 4 Finale, this information will be available.
Rainbow Dash is one of the main characters in My Little Pony: Friendship is Magic.
There is no known backstory for Queen Chrysalis as of yet, and it has not been confirmed if she was or wasn't Princess Papillon. Princess Papillion appears to have been a fan-made version of Chrysalis and so is not a cannon character. It is unlikely that Chrysalis has any connection with the Rainbow Kingdom.
In "Friendship is Magic", she does but in older "My Little Pony"s, the answer is no.