Why only two binary operations? Ring theory considers things with 2 operations and category theory 101 talks about products and coproducts.
I maybe understand why binary operations are more common to look at that trinary, quaternary, …, but why look at the interactions between only 2 of them? Surely the extension from 2 to many interlocking operations is not an obvious one…
 A: As usual, if you're confused about some general fact about categories, you should first see what it says about posets. In the context of posets products and coproducts correspond to two very natural operations, namely min and max. These are the two fundamental ways we might try to combine two elements of a poset to get another one, and roughly speaking the fact that there are two of them comes from the fact that posets have two "directions," namely "increasing" and "decreasing." 
More generally, a category has two "directions," namely "morphisms in" and "morphisms out." That's why, for example, a functor has two adjoints (when they exist), a left adjoint and a right adjoint, and that's it. (The relationship to limits and colimits is that limits and colimits can be described as adjoints.) 
When the poset is in particular a poset of subsets of a set, thought of as Boolean propositions, we can furthermore think of our two natural operations as and and or. 
To the extent that there's any connection to ring theory, it's because e.g. you can think of the primordial (semi)ring as being $\mathbb{N}$ with addition and multiplication, which in turn is a decategorification of the category of finite sets equipped with coproduct and product. But there are other ways to think about rings, e.g. as monoid objects in $(\text{Ab}, \otimes)$. From this perspective "two binary operations" is missing the point; we could replace $(\text{Ab}, \otimes)$ with an arbitrarily complicated monoidal category here. 
In any case, I dispute your claim that we only look at two binary operations. For example, a Poisson algebra has three. On the categorical side, a (bi)cartesian closed category also has three. The ring of symmetric functions has five. 
