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Basic theory of Abelian categories tells us that this is true if $\mathcal{D}$ is abelian. However, is this still true if we don't necessarily have kernels or cokernels?

Tag 05R4 in the stacks project (Derived Categories, Lemma 5.3) seems to implicitly assume that this is true.

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    A fellow student suggested using the fully faithful [Quillen embedding](https://ncatlab.org/nlab/show/Quillen+exact+category), considering $\mathcal D$ as an exact category in the trivial manner with only split exact sequences. My guess is this embedding won't preserve monic and epic morphisms? – 2017-02-16
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    @DustanLevenstein Right, The Quillen embedding for the trivial exact structure is, up to size issues, the Yoneda embedding into presheaves of abelian groups (every short exact sequence of representables splits since representables are projective.) And Yoneda preserves all monos, but essentially no epis, I guess only the split ones, for the same reason. – 2017-02-16
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    By the way, I don't see what you're referring to in Stacks, but in pre-triangulated categories a mono-epi is an iso, because all monos and epis are split. – 2017-02-20
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    Thanks. That's the information I needed. – 2017-02-20

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Nope. Topological abelian groups form an additive category, as can be seen directly or as a property of the category of abelian group objects in any category with finite products-a biproduct of $x,y$ in any category enriched over abelian groups is given by an object $z$ with morphisms $i_1,i_2:x,y\to z,p_1,p_2:z\to x,y$ such that $p_ji_j=1,p_0i_1=p_1i_0=0, i_1p_1+i_2p_2=1$, as follows from Yoneda and the same result for abelian groups, and if $C$ has finite products then we can construct such a diagram given abelian group objects $x,y$ with $z=x\times y$.

Anyway, it's just as easy to get nontrivial monic epics in AbTopGp as in Top. Consider the discrete and the indiscrete topologies on any abelian group, for instance, using that discrete and indiscrete spaces are closed under (finite) products.