equilibrium - Strong and Weak Acids/Bases

I'm so confused... $\ce{HClO}$ is a weak acid. then why is its conjugate base, $\ce{ClO-}$, a weak base in water? shouldn't they be inversely proportional? shouldn't $\ce{ClO-}$, be a strong base?

If $\ce{HClO}$ is a weak acid, that means it does not readily give up a proton and has a strong pull on them. So when it becomes a conjugate base, $\ce{ClO-}$, shouldn't it readily take protons and therefore be a strong base in water?

ALSO, when an acid is dissolved in water, and some of that acid dissociates into $\ce{H3O+}$, is the Kw of water negligible and you only consider the Ka when finding pH? But that confuses me because I thought you use 14, the exponent of Kw when finding pH.