On the definition of the direct sum and tensor product. Let's say I have two $R$-modules, $M$ and $N$. Here are two algebraic constructions I can form from $M$ and $N$.
1. Direct Sum
As a set we define the direct sum to be $M \oplus_R N = M \times N$ and we define our ring action and addition componentwise, i.e. $(a, b) + (c, d) = (a + c, b+d)$ and $r\cdot(a, b) = (r\cdot a, r\cdot b)$.
Now what I see frequently in textbooks is that authors will consider for instance $\alpha + \beta$ where $\alpha \in M$ and $\beta \in N$ as an 'actual sum' in $M \oplus_R N$, when technically $\alpha + \beta$ doesn't really make sense, since the actual summation that's taking place is $(\alpha, 0_N) + (0_M, \beta)$ and we are just identifying $(\alpha, 0_N) + (0_M, \beta)$ with $\alpha + \beta$.
This leads me to believe that $M \oplus_R N$ is really just a construction that allows us to add together elements from two different $R$-modules, and this was the purpose it was constructed for because once authors formally define $M \oplus_R N$ they quickly drop the formalism to use it in the way above, as a way to add elements of different $R$-modules together.
Now $M \oplus_R N$ also satisfies a universal property and I guess that the universal property is just a categorical restatement of the above. Please correct me if I'm wrong though.
2. Tensor Product
As a set we define $M \otimes_R N = A^{(M \times N)}/D$ where $A^{(M \times N)}$ is the free $R$-module on the set $M \times N$ and $D$ is the submodule of $M \times N$ generated by the relations
\begin{aligned}
    &(x+x', y) - (x, y) - (x', y) \\
   &(x, y+y') - (x, y) - (x, y')\\
    &(ax, y) - a\cdot(x, y) \\
&(x, ay) - a\cdot(x, y)
 \end{aligned}
and we define $x \otimes y = (x, y) + D \in M \times N / D$. Now as a consequence of our definition of $D$ in $M \otimes_R N$ in particular above, we end up having the following relations
\begin{aligned}
    &(x+x')\otimes y = x \otimes y + x' \otimes y \\
   &x\otimes (y + y') = x \otimes y + x \otimes y'\\
    &ax \otimes y = x \otimes ay = a(x \otimes y) \\
 \end{aligned}
and this holds for all $x, x' \in M$, $y, y' \in N$ and $a \in R$. Now the relations we end up with above look an awful lot like how we expect multiplication to behave (at least in $\mathbb{Z}$ as a prototype).
So this leads me to believe that the tensor product of two modules was constructed to allow us to 'multiply' together elements of the two individual modules.
And yes $M \otimes_R N$ has a universal property, that every map $f : M \times N \to P$ where $P$ is any other $R$-module factors uniquely through $M\otimes_R N$, and I've seen it useful to prove some properties of the tensor product, but it doesn't seem nearly as motivating to me to construct the tensor product as the argument for it being a construction that allows us to multiply together elements from individual modules.

Now onto my question(s).
From what I've seen so far of the direct sum and tensor product, it seems like they are formal constructions to solve simple problems, for the direct sum it's a construction that allows us to add together elements from different modules and for the tensor product it's a construction that allows us to multiply together elements from different modules.
Are these the intended purposes for which the direct sum and tensor product were created?
I understand that this could partially be true in the sense that maybe their universal properties are more desirable than what goes on intrinsically between elements within the construction. In other words it could be the case that how functions mapping into/out of $M \otimes_R N$ behave is more important than how actual multiplication between elements of $M \otimes_R N$ take place.
I know that this is the case analogously for quotient spaces in topology, where we sometimes care more about the universal property of quotient spaces than what the elements in the quotient space look like.
With that being said is it a naive viewpoint to view the tensor product and direct sum in the way in which I described (as constructions which allow us to add/multiply elements)?
Finally I think that in general one can define an object in a category by its universal property, so with that being said does an algebraic (intrinsic) definition of the direct sum or tensor product define the universal property  (extrnsic definition) associated with it and vice versa?
For example does the universal property of $M \otimes_R N$ define how we multiply elements within the tensor product, and does our definition of how elements are multiplied in $M \otimes_R N$ define its universal property?
 A: First off: everything you say above the questions is entirely correct. These absolutely are the origins of those things. I don't know of any interesting result that relies on the formal definition of the direct sum as an equivalence class of pairs (other than the equivalence of finite direct sums and products if you're using category theoretic definitions of those, which is easy to prove by just noting that this construction satisfies both), rather than the universal property versions (though the explicit construction of tensor products does crop up sometimes). The only thing that I'd add to your description is that they're not just any constructions that allows addition/multiplication of elements: they're the simplest possible constructions (in the sense captured by the universal properties). 
You are correct that in any nice-ish category, for any universal property, there is (up to isomorphism) exactly one universal object for that property: the proof is simple (applying the universal property both ways around gives you maps in both directions which turn out to be mutually inverse). Thus, you can take either the universal property or a particular construction of an object that satisfies that property as your base definition, and prove all of the same results, and you can use whichever you like at any point in a proof without any problems (there are even cases where there are multiple explicit constructions in common use, each of which is more convenient for using to prove certain results, with one of the main uses of the universal property being connecting all of these things together, so that you can just pick whichever is most convenient at the time). 

For example does the universal property of M⊗RN define how we multiply elements within the tensor product, and does our definition of how elements are multiplied in M⊗RN define its universal property?

Sort of. There are multiple explicit constructions of the tensor product, they're just all isomorphic (and the fact that they all satisfy the same universal property gives an easy proof of that). 
