Is there a surjective morphism $\widehat{\Bbb Z} \to \Bbb Z$?

In fact it seems there are no nontrivial group homomorphisms $\widehat{\mathbb{Z}} \to \mathbb{Z}$.

First let's note that there are no nontrivial group homomorphisms $\mathbb{Z}_{p} \to \mathbb{Z}$, since, as you say, the image of any such map must be $q$-divisible for $q \ne p$.

Write $\widehat{\mathbb{Z}} = \prod \mathbb{Z}_{p} = \mathbb{Z}_{2} \times \prod_{p \ne 2} \mathbb{Z}_{p}$. There are no nontrivial maps $\mathbb{Z}_{2} \to \mathbb{Z}$ by the above, and there are no nontrivial maps $\prod_{p \ne 2} \mathbb{Z}_{p} \to \mathbb{Z}$, since $\prod_{p \ne 2} \mathbb{Z}_{p}$ is $2$-divisible.