You can only dye leather armor, when you have a part of leather armor get the dye colour you want move the dye over the armor in your inventory and press X and then you have the armor in the dye colour you wanted. It's that easy!
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You can craft things with Lapis like: A Lapis Lazuli block, light blue dye, cyan dye, purple dye, magenta dye and blue wool. You can check out Minecraft Wiki for some help and tips.
No, but you can dye it's collar
Burn cactus in a furnace.
You cant unfortunately but there maybe a mod out there.
You can't. But with resource packs or mods you can.