I learned a lot from reading Bestvina and Feighn's article Notes on Sela's work: Limit groups and Makanin-Razborov diagrams. It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a background in geometric group theory (specifically, in understanding laminations on 2-complexes or group actions on $\mathbb{R}$-trees) to get the most out of it.

Regarding the question of whether or not the Hopf property is elementary: the answer is obviously no' if you allow infinitely generated examples. Indeed, consider the free group on countably many generators, $F_\infty$. The elementary theory is completely determined by the set of finitely generated subgroups, so $F_\infty$ is elementary equivalent to $F_2$. But $F_2$ is Hopfian and $F_\infty$ is not.

EDIT: I'm getting a little nervous about the claim that the elementary theory is determined by the list of finitely generated subgroups. However, Sela and Kharlampovich--Miasnikov proved that the natural inclusions $F_n\subseteq F_{n+1}$ are elementary embeddings, from which it does indeed follow that $F_\infty$ is elementarily equivalent to $F_2$.

I don't know a finitely generated example, although I agree that one must surely exist.

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I learned a lot from reading Bestvina and Feighn's article Notes on Sela's work: Limit groups and Makanin-Razborov diagrams. It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a background in geometric group theory (specifically, in understanding laminations on 2-complexes or group actions on $\mathbb{R}$-trees) to get the most out of it.

Regarding the question of whether or not the Hopf property is elementary: the answer is obviously no' if you allow infinitely generated examples. Indeed, consider the free group on countably many generators, $F_\infty$. The elementary theory is completely determined by the set of finitely generated subgroups, so $F_\infty$ is elementary equivalent to $F_2$. But $F_2$ is Hopfian and $F_\infty$ is not.

I don't know a finitely generated example, although I agree that one must surely exist.