I know that people who do repaints on Breyer horses often use colored spray primers (like black) as their base coat. In terms of the hair - it's probably synthetic and the ability to dye that stuff is hit or miss from what I've seen or tried. Some people have luck using sharpy pens to "dye" the hair, but you can risk it coming off and staining any doll that comes in contact with it. If possible your best bet is to remove the old hair and re-root the horse with the color you want.
As for taking them apart....good luck on that one, I don't think they were really designed for that but you might want to google something like "barbie horse disassemble"