Why are particular combinations of algebraic properties "richer" than others?

Examples!

Remember that most (if not all) abstract structures are motivated by specific examples. And it took a long time for the mathematicians to abstract from these examples and develop an axiomatic framework. Since you have been asking for groups: Permutation groups, Symmetry groups, Lie groups (aka transformation groups) and ideal class groups appeared naturally in the 19th century, even before the general notion of a group was born (Cayley, Galois, Klein, Kronecker, Lie, and many others). There is nothing interesting about the group axioms in themselves, but rather in the fact that they subsume what happens in so many examples, and that we can study many phenomena in specific examples for arbitrary groups. The same remarks apply - even more - to the notion of a category.

Monoids also appear very naturally in many examples. They have a rich theory, quite different from the theory of groups. But in general I would say monoids are harder to understand than groups. For example, whereas finitely generated commutative groups are classified, this is not the case for finitely generated commutative monoids. For this reason one often makes a monoid to a group by formally introducing inverses - this is called the Grothendieck group, which is especially important in K-theory.

Monoids even play a more important rule when we internalize them into arbitrary monoidal categories - this leads to the notion of a monoid object. Monoid objects in $\mathsf{Set}$ are monoids in the usual sense, but monoid objects in $\mathsf{Ab}$ are rings in the usual sense! This offers a considerable overlap between monoid theory and ring theory. In the commutative case, we can even go further and develop algebraic geometry for commutative monoid objects (Toen-Vaquié, Florian Marty).

I haven't worked with loops or near-rings, but I am pretty sure that these aren't covered in most lectures because there are not as many interesting examples as for groups and rings.

The conclusion is very simple: Abstract structures are motivated by specific examples. And this is not restricted to algebra. You could also go ahead and ask "why the union axiom in the definition of a topology?". The answer is the same: Because examples (especially the class of metric spaces) have motivated this axiom. Given a random system of operations and rules between them, you cannot really tell if this is interesting, natural, or not.

Since you already got 3 deep and pervasive answers, I want to concentrate on a single minor point.

I can ask what makes the specific combination of defining properties of a category so great?

In my humble opinion, almost nothing: you will certainly be interested in the work a friend of mine and I (in minor extent) are laying down where we study multi-object partial magmas and recover a great deal of classical "category" theory: you get these things called plots where your composition is not defined for each pair of consecutive arrows, and when it's defined it is possibly non-associative. Finally, you don't have identities everywhere.

"You fool, nothing good can come out of these poorly behaved, thorny things!"

:) not at all.

You can define "isomorphisms" (yes, without identities), and notice that "being an isomorphism" and "admitting an inverse" are different notions in this world, and that they collapse in the categorical world (a category is an associative plot, where the composition is defined and every object has a 1, in the same vein a monoid is an extremely smooth partial magma). You can then define isoids, i.e. plots where every arrow is an isomorphism.

We're even able to define morphisms of plots (p-unctors), natural transformations (trimmings, if I remember well the name Salvatore and I chose), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category) of plots to the category of associative plots, semicategories [in this case we've even two different adjunctions for two different fully faithful embeddings], and categories. Other things in the to-do list: what's a $n$-plot? How can one define localization of a plot with respect to a family of arrows? How about simplicial stuff, how about enrichments (whatever this means)?

"You are only children playing with symbols! Examples, Examples!!"

:) Functional analysis and symplectic geometry provide "natural factories" of examples of such structures. For example, one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.