Killing vector fields are those that verify $\mathcal{L}_X (g)=0$. This is equivalent to the following equation for a coordinate basis:
$$\nabla_a X_b + \nabla_b X_a=0$$
Do Killing vector fields satisfy $\nabla_a X^a + \nabla_b X^b=0$? I was wondering today if this happens but I wasn't able to prove or disprove it myself.
Your suggested equation reduces to $\nabla_a X^a = 0$ since $\nabla_a X^a = \nabla_b X^b$. Your suggested equation is also correct, since $$\nabla_a X^a = \nabla_a \left(g^{ab}X_b\right) = g^{ab}\nabla_a X_b$$ is a contraction of a symmetric tensor with an antisymmetric one.