Maintenance for the week of May 25:
• PC/Mac: No maintenance – May 25
• ESO Store and Account System for maintenance – May 27, 6:00AM EDT (10:00 UTC) - 4:00PM EDT (20:00 UTC)
No, you have to reach the alliance rank with a particular character to unlock the associated dyes, and ranks are not shared across characters -- even if they're the same alliance.
Once you unlock the dye, however, it will be available for all of your characters to use.