I would agree that it isn't particularly natural. Possibly 'what colours has it got on it?', or 'what colour combination is it?

If a subject is singular, its verb must be singular. If a subject is plural, its verb must also be plural - "is" is a singular verb. "What COLOUR IS it?" (singular) "What COLOURS does it have?" (plural) "What COLORS IS it?" (plural and singular incorrectly mixed) :)

You can, but it doesn't sound natural. "What colors?" sounds more common. Some people even say "What color is it?" even if the object is multi-colored.