In a pre-abelian category, it is known that the coimage and image of a morphism (if they exist) need not be isomorphic. Is there necessarily a bijection between them? For example, the the category of topological abelian groups, one can show that the canonical morphism $\varphi:\mathrm{coim}~f \to \mathrm{im}~f$ is identified with $f:X \to Y$ if $f$ is bijective. It's easy to cook up an example where this is not an isomorphism, but we already have a bijection.
If there is a bijection, it should be $\varphi$; but I'm having trouble playing around with the arrows without thinking of maps and sets. I don't see any reason why it should be a bijection, but I'm not too familiar with the exotic examples.
This is not true.
Let me first make a slightly pedantic point: the question doesn't make sense since there are pre-abelian (even abelian) categories which are not concretizable, and so we can't make sense of "bijection."
But even in the concrete case, the underlying sets need not be in bijection. In the very example you cite (topological abelian groups*), take the inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}$ (under addition, giving $\mathbb{Q}$ the subspace topology). The coimage is $\mathbb{Q}$ and the image is $\mathbb{R}$ (indeed, the map is the original inclusion, just as you mention), which are not in bijection.
*This should be Hausdorff TAGs -- thanks Arnaud D. for catching this.