From experience (woo premium!)
1) Horde, Roundcube, Squirrelmail all on.
2) If you mean the Backup Wizard in cPanel, yes. Otherwise you always could by exporting it in PHPMyAdmin and downloading it all via FTP.
3) Not sure how often its updated, but yes.
4) See above.
5) If by theme you mean cpanel theme, yes, otherwise please clarify the question.
6) Yes.
7) I have not attempted to use it, but the icon is there and accessing it doesn't give an error saying it's not enabled. I'm going to go with "more than likely is enabled, or we shouldn't have the button."
8) I know it's not mentioned but in case someone stumbles on this, Remote MySQL -IS- enabled on Premium, but not on Free. Just to tackle that one since it's been asked a lot before too