(Non-Hopfian) groups that only have quotients that are themselves or the trivial group.

group-theory axiom-of-choice hopfian-groups

I have provided a non-abelian example as a reply to the original MathSE question (where abelian was not discarded). (Sorry, I now realize I should maybe have replied here.)

This answers 1. As regards 2, as I said in a comment, there are no finitely generated examples regardless of the axiom of choice.

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