I've always thrown away weapons that deal double damage on Demons or Dragons, but I figured that I might have some use for it now that I'm reaching the later parts of the game. The question is, what counts as demons / dragons? Do only the red/frost/gold dragons count or does it affect all big lizards in the game?

There are devils and demons, and weapons affect both. As for dragons, I am not sure if the bonus extends to big lizards as well as the true dragons, but I suspect it will or at least that it will affect the flying drakes. However, from experience I have found that these weapons don't do that much extra damage. If you get the weapon in a skill you've mastered, then go ajhead and try it, but don't have your main fighter use a mace, say, instead of his sword, just because the mace does double. (Say if you have 1 or 4 ranks in mace and 8 or more in sword).

I find it's better to use magic against dragons, but if you are using knights it might be worth it. But like Larias mentioned, you should use weapons that you've trained in very well.

Demon/Dragon slaying weapons only double the actual weapon's damage - not the damage adding bonuses by:
- Might
- Heroism
- Weapon skill (Axe, Spear, Mace)

Mind you, they are still very effective and much better than other enchantments like the elemental damage bonuses.
But as others said, ALWAYS stick to the weapon type you specialized in.

because of this, dual wielded weapons, if both enchanted with demon slaying, litterally carve through The Hive.