Is a creatures ability a spell?

Creatures are spells while they are on the stack. However, if they have any triggered abilities as the result of being cast (such as Emrakul) or entering the battlefield (such as Ashen Rider) those are not spells; they are abilities.

Is a creature spell a creature?

A creature spell is any card with the Creature type. That is to say any creature when it is being cast (played) is a creature spell. Its easy when you start to think of spells as instants and sorceries, but in Magic every card you play (except lands) is a spell.