I know it is possible to change the color of sheep using dye. I've also learned that you can change the color of a tamed wolf this wasy too.
Does this also work on other mobs, specifically the Snowman?
No you can't dye a snow golem different colors. Here is a video to prove it:
Because you can't dye the snow blocks or the actual snow golem like I have demonstrated, the idea of a different coloration other than white is impossible,
As per the Minecraft wiki article on dyes and such, you can currently only dye wolves' collars and sheep.
Taken directly from the wiki:
Who knows what they may add in the future though. Pink horses?