ct.category theory - How cavalier can I be when demanding a category have direct sums?

I don't think your definition of direct sum is quite right (even if you add the obviously necessary condition that the isomorphisms be natural). My understanding is that a direct sum / biproduct is an object that is both a product and a coproduct in a compatible way. This is usually phrased by saying that you have coproducts and products, and the unique morphisms $0to 1$ and $Xsqcup Y to Xtimes Y$ are isomorphisms. In terms of your definition, I think this would be equivalent to saying that you have a zero object, and the composite isomorphism
$$hom(Z,Z) cong hom(X,X)times hom(Y,X)times hom(X,Y)times hom(Y,Y)$$
relates $1_Z$ to $(1_X,0,0,1_Y)$ (where $0$ is the map factoring through the zero object). It's true, but not (I think) obvious, that if you have products and coproducts and an arbitrary natural family of isomorphisms $Xsqcup Y cong Xtimes Y$, then you actually have biproducts. But I don't think this works as a definition for an individual biproduct.

As to your actual question, I don't have a complete answer, but one thing to note is that in the world of categories enriched over additive monoids (or groups), direct sums are absolute (co)limits, aka Cauchy (co)limits. That means that they are automatically preserved by any AbMon-enriched functor, and moreover the 2-category of AbMon-enriched categories with direct sums is reflective in the 2-category of all AbMon-enriched categories. Therefore, after performing any "free" or "quotient" or "colimit" construction on AbMon-enriched categories, you can always apply the reflector to add any direct sums that might be missing (and whatever direct sums you might already have had won't be changed). In particular, this provides a construction of an additive category "presented" by any notion of generators and relations: first generate the free AbGp-enriched category, then reflect into additive categories.

In general, it's not obvious to me that if you add some additional structure freely (like kernels or cokernels), then apply the above "Cauchy-completion" reflector, that the presence of the new thing you added is preserved by the reflector. But if it isn't, then perhaps some sort of sequential colimit of successive approximations could be performed. Note that of the other constructions you mentioned, splitting of idempotents is also an absolute (co)limit, so it behaves similarly to direct sums, whereas kernels and cokernels are not.

However, none of this really answers the question you actually asked, which is whether such "free" constructions in the world of AbGp-enrichment already preserve the presence of direct sums, without the need to Cauchy-complete. I would guess that in general they don't, but I don't have a counterexample.