Let me address the first part of the question about splitiings. (To address the second one, one could probably write another post of similar length
Relevant notions are flatness of a module and pure-exactness of a short exact sequence.
I suggest the following point of view.
Let's look at the category $R\operatorname{-Mod}$ of left $R$-modules, and the category of short exact sequences $R\operatorname{-SES}$ of left $R$-modules (where morphisms are commutative 2 by 3 boxes).
$\operatorname{Hom}_R(-, -)$ defines two "pairings" between those two categories with values in 3-term complexes of abelian groups. First pairing corresponds to the case where the left argument is a SES, and right is the module; the second one is reverse of that.
One can wonder: what is captured by "orthogonality" with respect to such pairing, where we deem a pair to be orthogonal if it is sent to an exact complex of abelian groups?
(For the module this orthogonality is exactness of (co)representable functor, and for SES it corresponds to "intrinsic exactness" from the POV of Hom-functor.)
One easily notes that modules which are left orthogonal to every SES are exactly projective ones; moduels which are right orthogonal to every SES are injective; and SES which are orthogonal to any module are exactly the split ones.
Last one is not so trivial when SES is the left argument; you need an injective module $W$ with the property that $\operatorname{Hom}_R(-, W)$ is faithful, but those always exist, as the products of injective envelopes of all simple $R$-modules.
Now let's turn to the situation of a pairing $- \otimes_R -$ between $R\operatorname{-SES}$ and $\operatorname{Mod-}R$.
In this case left orthogonal-to-everything modules are called flat, and right orthogonal-to-everything sequences are called pure exact. Two morphisms in a pure SES are called pure monomorhism and pure epimorphism correspondingly.
They behave quite similarly to the class of usual monomorphisms and epimorphisms in category of modules. Pure monomorphisms are closed under composition, products, pushouts along arbitrary morphisms; if a composition $A \to B \to C$ is pure, then $A \to B$ is pure. Unlike monomorphisms, pullback of pure monomorphism is not necessarily pure! For example, $(2, 0, 0)$ — intersection of two direct summands $(1, 1, 0)$ and $(1, 0 , 1)$ in $\Bbb Z/4 \oplus \Bbb Z/2 \oplus \Bbb Z/2$ is not purely embedded.
There are various equivalent criteria of flatness and pure-exactness. The most illuminating one (if you are familiar with basic category thery) is is the following.
Recall that a colimit is called filtered (directed), if the indexing diagram is filtered, i. e. has a cone over every finite subdiagram (if the diagram is a poset with finite joins). Every filtered diagram admits a cofinal functor from a directed one, so. in a sense, filtered colimits reduce to directed ones; but filtered have the advantage of being more flexible. For example, the category $R\operatorname{-mod}/M$ of morphisms from (all) finitely presented modules to a fixed module $M$ is always filtered, but seldom directed.
Basic fact is that every module is a filtered colimit of finitely presented ones; more precisely, it's a colimit of a functor $R\operatorname{-mod}/M \to R\operatorname{-Mod}$ sending a morphism $f: C \to M$ to $C$.
Lemma (Lambek). Module is flat iff it is a directed colimit of projective (or, equivalently, free, or finitely presented projective, or finitely generated free) modules. Class of flat modules is closed under filtered colimits.
Dual lemma (Raynaud?). SES is pure iff it is a directed colimit of split SES. (Alternatevely: monomorphism is pure iff it is a directed colimit of split monomorphisms.) Class of pure monomorphisms is closed under filtered colimits.
Note that flat module is not necessarily projective. Abelian group ($\Bbb Z$-module) is flat iff it is torsion-free, and projective iff it is free, so $\Bbb Q$ is a non-projective flat $\Bbb Z$-module.
Similarly, pure morphisms are not necessarily split. Inclusion of torsion subgroup into any abelian group $tA \to A$ is always pure, but if $A = \prod_{p \text{ prime}} \Bbb Z/p$, then $tA$ is not a direct summand (explicit description of torsion subgroup is a bit delicate). Moreover, for any set of $R$-modules $\{M_i\}_{i \in I}$ the canonical morphism $\bigoplus_I M_i \to \prod_I M_i$ is pure, but not necessarily split. Countable sum of copies of $\Bbb Z$ is not a direct summand of countable product; this can be derived from Specker's proof that $\prod_{\Bbb N} \Bbb Z$ is not free.
There are a few other properties of a SES of left $R$-modules $$M \xrightarrow{f} N \xrightarrow{g} K,$$ equivalent to being pure.
- $\operatorname{Hom}_{\Bbb Z}(f, \Bbb Q/\Bbb Z): \operatorname{Hom}_{\Bbb Z}(N, \Bbb Q/\Bbb Z) \to \operatorname{Hom}_{\Bbb Z}(M, \Bbb Q/\Bbb Z)$ is a split epimorphism of right $R$-modules.
- For every finitely presented $C$ the map $f_*: \operatorname{Hom}_{R}(C, N) \to \operatorname{Hom}_{R}(C, K)$ is a surjection. (In other words, it is a SES which is right Hom-orthogonal to finitely left $R$-modules. This ties purity to the interpretation of projectivity as "orthogonality" condition.)
- For every commutative square below
$$\require{AMScd}
\begin{CD}
R^k @>{t}>> R^l\\
@V{q}VV @VV{p}V \\
M @>{f}>> N
\end{CD}$$
there's a map $s: R^l \to M$, such that $q = st$.
- For every commutative square below, where $C, C'$ are finitely presented
$$\require{AMScd}
\begin{CD}
C @>{t}>> C'\\
@V{q}VV @VV{p}V \\
M @>{f}>> N
\end{CD}$$
there's a map $s: C' \to M$, such that $q = st$.
Analogously, the following properties of a right $R$-module $X$ are equivalent to being flat.
- Left $R$-module $\operatorname{Hom}_{\Bbb Z}(X, \Bbb Q/\Bbb Z)$ is injective.
- For every finitely presented $C$, and every surjection $\pi: Y \to X$ the map $\pi_*: \operatorname{Hom}_{R}(C, Y) \to \operatorname{Hom}_{R}(C, X)$ is a surjection. (In other words: every SES ending in $X$ is pure.)
- If $r \in R^k$ is in the kernel of a map $f: R^k \to X$, then there's a factorisation of $f$ as $f'g: R^k \xrightarrow{g} R^l \xrightarrow{f'} X$, such that $r$ lies in the kernel of $g$ already.
- Every morphism $f: C \to X$ from a finitely presented module $C$ has a factorisation through a free module.
Notion of purity appears in various parts of algebra. One of its noticeable applications is the following
Theorem (Olivier '70, Mesablishvili '98). The following conditions are equivalent for the morphism of commutative rings $\phi: S \to T$:
- $\phi$ is a pure monomorphism of $S$-modules (i. e. $M \otimes \phi: M = M \otimes_S S \to M \otimes_S T$ is mono for every $S$-module $M$);
- $A \otimes \phi: A = A \otimes_S S \to A \otimes_S T$ is mono for every $S$-algebra $A$
- $- \otimes_S T: \operatorname{Mod-}S \to \operatorname{Mod-}T$ is faithful;
- $- \otimes_S T: \operatorname{Mod-}S \to \operatorname{Mod-}T$ reflects isomorphisms;
- $- \otimes_S T: \mathrm{CRing}/S \to \mathrm{CRing}/T$ reflects isomorphisms.
Related to this is another (very nontrivial)
Theorem (Raynaud, Gruson '71, Angermüller '21). If a morphism of commutative rings $\phi: S \to T$ is pure, then $P$ is a projective $S$-module iff $P \otimes_S T$ is a projective $T$-module. (In particular, it holds if $\phi$ is faithfully flat.)
Other incarnations of purity arise in model-theoretic methods of representation theory, and first order theory of modules. There's a wonderful book by Mike Prest called Purity, Spectra and Localisation, 2009 on this topic.
To broaden the scope, definition 4 of purity can be modified to make sense in any category with filtered colimits. There's (essentially the only readable) treatise of the topic in J. Adamek, J. Rosicky, Locally presentable and accessible categories, 1994. It happens that filtered colimits — or, rather, preservation of those by applying Hom(C, -) — are exactly the right tool to measure "smallness" of an object C (at least in ones where filtered colimits exist); the theory coming out of this observation is quite beautiful and useful.
