The answer to the first question is yes. If A and B have a direct sum A ⊕ B in C, then there are inclusions iA : A → A ⊕ B, iB : B → A ⊕ B and projections pA : A ⊕ B → A, pB : A ⊕ B → B such that pAiA = 1, pBiB = 1, and iApA + iBpB = 1. Conversely, the existence of such maps in an Ab-enriched category make A ⊕ B a direct sum of A and B, even if we do not asssume a priori that A and B have a direct sum. Now if we form the quotient C/I by an ideal I, and two objects A and B with a direct sum A ⊕ B in C, the image of this system of maps presents the image of A ⊕ B as the direct sum of the images of A and B. In short, direct sums are absolute colimits, and as the quotient functor C → C/I is essentially surjective (indeed, bijective on objects), every pair of objects of C/I inherits a direct sum from C.
Observe that $\bigoplus_I : \textbf{Ab}^I \to \textbf{Ab}$ is a conservative exact functor: it is right exact by general nonsense, it preserves monomorphisms (because e.g. $\bigoplus_{i \in I} A_i$ is naturally a subgroup of $\prod_{i \in I} A_i$), and it is conservative because $\bigoplus_{i \in I} A_i \cong 0$ implies each $A_i \cong 0$.
It seems to me that you are looking for a conservative exact functor $\mathcal{A} \to \textbf{Ab}$, so this observation implies that allowing $I$ with more than one element does not add any generality.
For small abelian categories $\mathcal{A}$, the Freyd–Mitchell embedding theorem gives a fully faithful exact functor $\mathcal{A} \to R\textbf{-Mod}$ (where $R$ is a not necessarily commutative ring), so every small abelian category admits a conservative exact functor $\mathcal{A} \to \textbf{Ab}$.
In particular, $\mathcal{A}$ will be isomorphic to a (not necessarily full) abelian subcategory of $\textbf{Ab}$, by which I mean a subcategory that is closed under finite direct sums/products, kernels, and cokernels.
If $\mathcal{A}$ is not required to be small then one has to do more work.
The existence of a conservative exact functor $\mathcal{A} \to \textbf{Ab}$ is itself a size restriction on $\mathcal{A}$.
For example:
Proposition.
If there is a conservative exact functor $\mathcal{A} \to \textbf{Ab}$, then $\mathcal{A}$ is locally small and wellpowered.
Proof.
A conservative exact functor is automatically faithful, and $\textbf{Ab}$ is locally small, so $\mathcal{A}$ must also be locally small.
Similarly, a conservative exact functor induces embeddings of subobject lattices, and $\textbf{Ab}$ is wellpowered, so $\mathcal{A}$ must also be wellpowered. ◼
Maybe you don't believe in categories that are not locally small and wellpowered.
Even so, I am not aware of any embedding theorems that work for arbitrary locally small and wellpowered abelian categories.
(I am also not aware of counterexamples.
Perhaps there is some large cardinal axiom that implies it can be done.)
In practice, it is not necessary to embed the entire category to prove the theorems you want – you can usually find a small abelian subcategory containing all the objects and morphisms you need for your theorem and then you can embed that subcategory.
Asking for an embedding of the whole category at once is being greedy – like asking for a global holomorphic chart of a complex manifold when local charts suffice.
Here are some embedding theorems I know for non-small abelian categories.
Theorem.
If $\mathcal{A}$ is a Grothendieck abelian category then there is a conservative exact functor $\mathcal{A}^\textrm{op} \to \textbf{Ab}$ that has a left adjoint.
Proof.
It is a well-known theorem that Grothendieck abelian categories have injective cogenerators.
But an injective cogenerator of $\mathcal{A}$ is precisely an object $I$ such that $\textrm{Hom}_\mathcal{A} (-, I) : \mathcal{A}^\textrm{op} \to \textbf{Ab}$ is a conservative exact functor.
Furthermore, representable functors in cocomplete categories automatically have a left adjoint, and $\mathcal{A}^\textrm{op}$ is indeed cocomplete. ◼
Maybe contravariance is jarring.
But $\textbf{Ab}$ is itself a Grothendieck abelian category, so the theorem (or Pontryagin duality) gives us a conservative exact functor $\textbf{Ab}^\textrm{op} \to \textbf{Ab}$, and composing them yields a conservative exact functor $\mathcal{A} \to \textbf{Ab}$.
The following is a small generalisation.
Theorem.
If $\mathcal{A}$ is a locally small abelian category and has a small generating set, then there is a conservative exact functor $\mathcal{A}^\textrm{op} \to \textbf{Ab}$.
Proof.
The hypothesis implies there is a small full abelian subcategory $\mathcal{B}$ containing the given small generating set of $\mathcal{A}$.
We get a fully faithful functor $\mathcal{A} \to [\mathcal{B}^\textrm{op}, \textbf{Ab}]$, but in any case it is not automatically exact.
Let $\textbf{Lex} (\mathcal{B}, \textbf{Ab})$ be the full subcategory of left exact functors $\mathcal{B}^\textrm{op} \to \textbf{Ab}$.
Then, the earlier functor factors as a fully faithful exact functor $\mathcal{A} \to \textbf{Lex} (\mathcal{B}, \textbf{Ab})$.
But $\textbf{Lex} (\mathcal{B}, \textbf{Ab})$ is a Grothendieck abelian category, so we may apply the earlier theorem to conclude. ◼
Best Answer
I don't know if this is going to answer your question but here's some relevant background. Splitting idempotents has a very special property from a categorical point of view: it is an absolute colimit (and also an absolute limit), meaning that it is preserved by any functor whatsoever. Two somewhat more familiar examples of absolute colimits, in enriched settings:
In general we can ask for the completion of a (possibly enriched) category under absolute colimits; this is called its Cauchy completion because it turns out to specialize to Cauchy completion when thinking of metric spaces as enriched categories. We have:
In particular, if we consider a ring $R$ as a one-object linear category $BR$, its Cauchy completion is exactly the category of finitely generated projective $R$-modules, which we famously use to define K-theory. This construction is a complete Morita invariant in the following sense: two rings $R, S$ are Morita equivalent (meaning $\text{Mod}(R) \cong \text{Mod}(S)$) iff their categories of finitely generated projective modules are equivalent (and this generalizes to enriched categories and Cauchy completions).
Cauchy completion is in some sense the "most harmless" and "most inevitable" completion: if you are ever going to apply a functor from your category $C$ to a category $D$ with colimits then every absolute colimit of objects in $C$ will appear in $D$ anyway (all idempotents will be split, etc.) so you might as well add them in first. Unlike the Yoneda embedding, which is the free cocompletion, Cauchy completion does not destroy colimits that may already exist in $C$. And because absolute colimits are preserved by all Hom functors $\text{Hom}(c, -)$, unlike adjoining colimits in general, the morphisms both into and out of an absolute colimit are already uniquely determined, so you have no choice how to do it anyway.
On the other hand attempting to write down some sort of completion of a linear category producing an abelian category seems quite tricky and potentially unwieldy. We can take the free cocompletion ($\text{Ab}$-valued presheaves) but again this destroys most existing colimits. Maybe the Isbell envelope has better properties but I don't know anything about it. Meanwhile the Cauchy completion is relatively easy to work with.