= Solution
An <abelian category> is an <additive category> with <kernels in a category> and <cokernels in a category> in which the canonical comparison from <coimage> to image of every morphism is an <isomorphism>. This is the standard image-coimage form of the definition, equivalently expressed by normality of every <monomorphism> and <epimorphism>. Additivity means abelian-group hom-sets with bilinear composition, a <zero object> and finite <biproducts>. A sequence is exact at an object when the image of the incoming morphism is the <kernel in a category> of the outgoing one.
For $f:X\to Y$, construct $q:X\to\operatorname{coim}f=\operatorname{coker}(\ker f)$ and $j:\operatorname{im}f=\operatorname{ker}(\operatorname{coker}f)\to Y$. The <universal properties> give $f=j\bar f q$: first $f$ kills its <kernel in a category> and factors through $q$; then the resulting map is killed by $\operatorname{coker}f$ because $q$ is epic, so it factors through $j$. The comparison $\bar f$ is invertible by the abelian axiom. <Cokernels in a category> are epic and <kernels in a category> monic, directly by uniqueness in their <universal properties>, so $e=\bar f q$ is epic and $j$ monic. Thus $\boxed{f=je\text{ is an epimorphism-monomorphism factorization}}$.
In a short exact sequence $0\to X\overset{i}{\to}Y\overset{p}{\to}Z\to0$, we have $i=\ker p$ and $p=\operatorname{coker}i$, up to the canonical <isomorphisms>. If $r:Y\to X$ retracts $i$, the map $1_Y-ir$ kills $i$ and factors as $sp$ for some $s:Z\to Y$. Composing with $p$ gives $psp=p$, so epicity gives $ps=1_Z$. Conversely if $ps=1_Z$, then $1_Y-sp$ is killed by $p$ and factors as $ir$. Composing with $i$ gives $iri=i$, so monicity gives $ri=1_X$. Consequently
$$
\boxed{i\text{ is split monic}\iff p\text{ is split epic}.}
$$
In that case $Y\cong X\oplus Z$.
For the derived-functor requests, use additive <right exact functors>, as customary in abelian homological algebra. An <abelian category with enough projectives> permits an augmented <projective resolution> $\cdots\to P_2\to P_1\to P_0\to X\to0$, obtained by repeatedly taking an <epimorphism> from a projective onto the previous <kernel in a category>. Apply $F$ and define
$$
L_nF(X)=H_n(FP_\bullet),\qquad L_0F(X)\cong F(X).
$$
Projective lifts construct maps of resolutions; the comparison and chain-homotopy results make the construction independent of choices and functorial. For a short exact sequence, the horseshoe construction produces a degreewise split short exact sequence of resolutions, hence of their images under additive $F$. Its <homology objects> give the long exact sequence
$$
\cdots\to L_2F(X)\to L_2F(Y)\to L_2F(Z)\to
L_1F(X)\to L_1F(Y)\to L_1F(Z)\to
F(X)\to F(Y)\to F(Z)\to0.
$$
This supplies the requested sketch; detailed comparison and functoriality proofs are not needed here.
A projective object $X$ has a resolution concentrated in degree zero, so $L_1F(X)=0$ for every such $F$. Conversely choose $0\to K\overset{i}{\to}P\to X\to0$ with $P$ projective, and take $F=\mathcal A(-,K)$ with target $\mathbf{Ab}^{\mathrm{op}}$. Precomposition makes this covariant to the <opposite category>. The usual left exactness of contravariant Hom, which follows directly from the <cokernel in a category> <universal property>, says exactly that this $F$ is <right exact> in that opposite target. If $L_1F(X)=0$, the long exact sequence and $L_1F(P)=0$ make $F(K)\to F(P)$ monic in $\mathbf{Ab}^{\mathrm{op}}$. Equivalently,
$$
\operatorname{Hom}(P,K)\longrightarrow\operatorname{Hom}(K,K),\qquad r\longmapsto ri,
$$
is epic, hence surjective, in <abelian groups>. The identity of $K$ has a preimage, giving a retraction of $i$. The sequence splits, and $X$ is a <direct summand> of $P$, hence projective: a lifting problem for $X$ extends along the projection $P\to X$, is solved for $P$, and restricts along the summand inclusion. Therefore
$$
\boxed{X\text{ projective}\iff L_1F(X)=0\text{ for every additive right exact }F.}
$$
This is <projectivity detected by all first left derived functors>. The following parts prove the three requested equivalent formulations of a <hereditary abelian category with enough projectives>.
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