Which comes first, damage or animation?

They try and do it by animation. I'm nearly positive that animation comes first then damage.

depends entirely on the ability or champ.

But more often than not the animation comes before the damage but even if you try to dodge the damage after a certain frame of the animation you are guaranteed to receive the damage even if you teleport across the map unless its particular ability types or attack types.

Older champs are more likely to have cases where the damage actually comes before the animation. Newer champs are cleaner designs.