Eventually, with these axioms the OP's E-structures are known as quasigroups and the W-structures are known as medial quasigroups (thanks to user zeb for the terminology). The Bruck–Murdoch-Toyoda theorem (same link, also mentioned by user zeb) says that a medial quasigroup is always of the form $(A,\ast)$ defined from an abelian group $(A,+)$, for some commuting pair of automorphisms $(\varphi,\psi)$ of $A$ and constant $c\in A$, with $x\ast y=\varphi(x)+\psi(y)+c$.

(This is quite restrictive: for instance, it implies that for any two $x,x'$, writing $L_x(y)=xy$, we have $L_x^{-1}L_{x'}(y)=y+\psi^{-1}(\varphi(x)-\varphi(x'))$, so that $L_{x}^{-1}L_{x'}$ is a translation. In particular, if $A$ is finite with $|A|\notin 4\mathbf{Z}+2$, all permutations $L_x$ have the same signature.) Similarly for right $\ast$-multiplications.

I think one can clarify this in a clear-cut way. Partially following OP's terminology, I'll call E-structure a set $X$ endowed with three binary laws $\sigma=\cdot$, $\lambda$, $\rho$ satisfying the given axioms 1,2. I'll write $\sigma(x,y)=xy$. It is a W-structure if moreover it satisfies the 3rd axiom (depending only on $\sigma$), namely $(xy)(zt)=(xz)(yt)$.

Let me now consider this as a magma $(X,\sigma)$ and then discuss existence and uniqueness of $\lambda,\rho$.

A first remark is that in the presence of a unit, Axiom 3 implies commutativity. Also (commutativity + associativity) implies Axiom 3, so Axiom 3 for a monoid means commutative.

The existence of $\lambda$ means that $(xy,y)=(x'y',y')$ implies $x=x'$. That is, $xy=x'y$ implies $x=x'$. This just means that the magma $(X,\cdot)$ is right-cancelative. Similarly the existence of $\rho$ means left-cancelative, and

For a magma $(X,\cdot)$, there exists an E-structure with $\sigma$-law is the magma product $\cdot$ iff $(X,\cdot)$ is cancelative. (And it is an W-structure iff it satisfies identically $(xy)(zt)=(xz)(yt)$.

About uniqueness: clearly $\lambda$ is uniquely determined on the image of the map $X^2\to X^2$, $(x,y)\mapsto (xy,y)$, and can be arbitrarily modified on the complement of this image $I=\bigcup_{y}Xy\times\{y\}$. A similar thing happens for $\rho$, with $J=\bigcup_x\{x\}\times xX$. So an E-structure (resp. W-structure) is a cancelative magma, along with some choice of maps on these complements (which seem not to be the point of interest, as OP focusses on the law $\cdot=\sigma$). (Nevertheless, modifying $\lambda$ and $\rho$ might affect direct indecomposability.)

Actually $I=X^2$ iff $Xy=X$ for all $y$, etc. Hence, the pair $(\lambda,\rho)$ is at most unique if and only if all left and right multiplications in the magma $X$ are surjective. As seen above, its existence means they're injective. So the existence and uniqueness of $(\lambda,\rho)$ means that left and right multiplications are bijective. (For a semigroup, this means being a group.)

OP's theorem, if I understand correctly, can be restated as: if $(X,\cdot)$ is a cancelative commutative idempotent magma, then it admits a structure of $\mathbf{Z}[1/2]$-module such that $xy=\frac12(x+y)$ for all $x,y$.

I think one can clarify this in a clear-cut way. Partially following OP's terminology, I'll call E-structure a set $X$ endowed with three binary laws $\sigma=\cdot$, $\lambda$, $\rho$ satisfying the given axioms 1,2. I'll write $\sigma(x,y)=xy$. It is a W-structure if moreover it satisfies the 3rd axiom (depending only on $\sigma$), namely $(xy)(zt)=(xz)(yt)$.

Let me now consider this as a magma $(X,\sigma)$ and then discuss existence and uniqueness of $\lambda,\rho$.

A first remark is that in the presence of a unit, Axiom 3 implies commutativity. Also (commutativity + associativity) implies Axiom 3, so Axiom 3 for a monoid means commutative.

The existence of $\lambda$ means that $(xy,y)=(x'y',y')$ implies $x=x'$. That is, $xy=x'y$ implies $x=x'$. This just means that the magma $(X,\cdot)$ is right-cancelative. Similarly the existence of $\rho$ means left-cancelative, and

For a magma $(X,\cdot)$, there exists an E-structure with $\sigma$-law is the magma product $\cdot$ iff $(X,\cdot)$ is cancelative. (And it is an W-structure iff it satisfies identically $(xy)(zt)=(xz)(yt)$.

About uniqueness: clearly $\lambda$ is uniquely determined on the image of the map $X^2\to X^2$, $(x,y)\mapsto (xy,y)$, and can be arbitrarily modified on the complement of this image $I=\bigcup_{y}Xy\times\{y\}$. A similar thing happens for $\rho$, with $J=\bigcup_x\{x\}\times xX$. So an E-structure (resp. W-structure) is a cancelative magma, along with some choice of maps on these complements (which seem not to be the point of interest, as OP focusses on the law $\cdot=\sigma$). (Nevertheless, modifying $\lambda$ and $\rho$ might affect direct indecomposability.)

Actually $I=X^2$ iff $Xy=X$ for all $y$, etc. Hence, the pair $(\lambda,\rho)$ is at most unique if and only if all left and right multiplications in the magma $X$ are surjective. As seen above, its existence means they're injective. So the existence and uniqueness of $(\lambda,\rho)$ means that left and right multiplications are bijective. (For a semigroup, this means being a group.)

OP's theorem, as stated in the post can be restated as: if $(X,\cdot)$ is a cancelative commutative idempotent magma, then it admits a structure of $\mathbf{Z}[1/2]$-module such that $xy=\frac12(x+y)$ for all $x,y$. [This is precisely equivalent to the formulation in the post. OP claims in a comment that the result is weaker, namely that every cancelative commutative idempotent magma embeds into another one that admits such a structure of $\mathbf{Z}[1/2]$-module inducing the magma law as average map.]