Answer 1. Take the projective line $\mathbb{P}^{1}_{k}$, and let $p$ be the correspondence projecting to a point. Then this has an image, namely $\mathrm{Spec}(k)$. However, it does not have a kernel. One can see this by looking at cohomological realisations. If the kernel existed, it would only have an $\mathrm{H}^{2}$; since the $\mathrm{H}^{0}$ is accounted for by $\mathrm{Spec}(k)$.
Answer 2. Since the Hom-sets have $\mathbb{Q}$-coefficients, every isogeny becomes an isomorphism. But maybe you find this answer a bit cheating.