Do color dyes give specific properties?
When I go to purchase a color dye, in the detail section of the color dye there is stuff written like "This pigment boost speeds" etc.
Do color dyes really have any impact on the character properties (other than appearance)?
Dyes only change the aesthetic look of your character.
That said, there are achievements directly linked to having a full set of certain colours equipped, and taking out certain bosses.
Dyes only change the color of a piece of armor. The text in the detail section is just flavor.