One algebraic way to motivate this is to observe that the signs in the differential for the Hom are precisely what is needed for 0-cycles in the $\hom(A,B)$ complex to be the set of morphisms of complexes $A\to B$ (and also, that the 0th homology group $H_0(\hom(A,B))$ is the set of homotopy classes of morphisms $A\to B$). This is quite great.
Once you decide you want this, all the other signs you mention follow because you need various things to hold. For example, you want the adjuntion between $\hom$ and $\otimes$ to hold for the internal versions, so this forces you to add signs to the $\otimes$, and so on.
A topological, indirect explanation for the appearence of most signs is that in the long exact sequence of bases maps corresponding to a map $f:X\to Y$, which looks like
$$X\to Y\to C(f)\to S(X) \to S(Y) \to S(C(f)) \to \cdots$$ there is a "sign" which you cannot get rid of. This sign reproduces itself in every algebraic version.
Answer 1. The closest thing to this construction I have seen is the Eilenberg-Watts theorem, which says that for any right exact functor $F\colon R$-mod$\to Ab$ that commutes with arbitrary direct sums, we have a natural isomorphism $F(-)\cong F(R)\otimes_R-$, where $F(R)$ is given its natural structure as a right $R$-module.
The key observation to Eilenberg's original proof is that given an $R$-module $M$, the canonical module homomorphism $\bigoplus_{m\in |M|}R\twoheadrightarrow M$ is in fact an $R$-bilinear function when considered as a two-variable function, and that consequently so is the image $\bigoplus_{m\in|M|}F(R)\twoheadrightarrow F(M)$ of the map under $F$. Then a little bit of diagram chasing shows that the induced map $F(R)\otimes_R M\to F(M)$ is in fact an isomorphism.
Hence, you can obtain constructions of the tensor product $M\otimes_R N$ from any right exact, direct-sum preserving functor $F_M$ for which $F_M(R)=M$. Thus, one should not be surprised at there being a ton of different constructions of the tensor product.
It is not the result of the theorem that's relevant here, however, but rather the idea behind the proof. Adapting, it seems to boil down to the observation that $\bigoplus_{m\in|M|} N$ has $|M|_{Ab}\otimes_\mathbb Z N$ as the natural quotient by the additive relations of $|M|_{Ab}$ that $|M|_{Set}$ has forgotten, and that $\bigoplus_{n\in|N|} M$ has $M\otimes_\mathbb Z |N|_{Ab}$ as the natural quotient by the additive realtions of $|N|_{Ab}$ forgotten by $|N|_{Set}$ (the proofs of these facts should be the same as in Eilenberg's proof). Then all your construction does is realize $M\otimes_R N$ as the pushout of the two.
Answer 2. I do not know of a textbook that does this stuff.
Answer 3. What is wrong with free modules? Your distaste for them mystifies me since I perceive algebraic objects are by any reasonable definition algebraic by virtue of being given as quotients of free objects (that's what an equation is). If I were teaching, what I would do is show how the (classical) explicit construction of the tensor product is nothing more than expanding the definitional hom-tensor adjunction, the internal hom to hom-set relationship, and the cartesian product to set-hom adjunction. Since I do not think the explicit construction is ever helpful for computational purposes, I would focus on the categorical properties from which one can both deduce the construction, and actually use for computation (e.g. preservation of direct sums and right exactness).
Best Answer
Let $(C,d_C)$ and $(D, d_D)$ be two chain complexes of $R$-modules, where $d_C$ and $d_D$ are differentials of degree $+1$. By definition, each $C_i$ and $D_j$ are $R$-modules. We want to "compose" the above complexes in a tensorial way; the definition you propose has 2 main effects:
is compatible with the (co)homological grading. In fact, for all $c\in C_i$ and $g\in D_j$, s.t. $i+j = n$, i.e. $c\otimes_R g\in (C\otimes_R D)_{n}$ then
$$d_{C\otimes_R D}(c\otimes_R g)= d_C c\otimes_R g + (-1)^i c \otimes_R d_Dg\in (C\otimes_R D)_{n+1}, $$
as $d_C c\in C_{i+1}$ and $d_D g\in D_{j+1}$. We used the Koszul sign rule.
For topological / geometric insights I refer to the text "Rational Homotopy Theory" by Felix, Halperin and Thomas. For formal definitions and applications in homological algebra the book by Gelfand and Manin is recommended, instead.