1. I'd say that a lot of "higher-dimensional mathematics" concerns spaces with operations of arbitrary finite arity. I'm thinking of things like planar algebras, operads, ...
   
   Let me mention one area that I like, which are various things I'd call "associative", and rather than trying to give precise definitions I'll mention planar algebras. A planar algebra includes in the data a $k$-ary operation for every way to draw nonintersecting curves (that are either closed or end on a boundary) on a disk minus $k$ subdisks. These operations are required to compose in any way that you can stick a disk-minus-holes into a hole in another disk (with the requirement that any curves ending on the glued-along boundary components match up). Then there's also an associativity requirement that says that everything only depends on the topology of the diagram, not the geometry.
   
   Anyway, it is possible to write any "planar operation" as a composition of binary operations (although you need infinitely many "basic" binary operations), but this is the wrong way to think about it, I claim. In particular, there's really no canonical choice how to write something as a composition of binaries.

2. From this point of view, let's now revisit usual associative multiplication. The associativity says nothing more nor less than: ab c = a bc. Drawn this way, it's clear that this is again a statement that "only the geometry matters, not the topology". But the point is that the usual multiplication is "one-dimensional", in that the ambiant space where things like "a", "b", "c" are put is a line. (Compare planar algebras, which are inherently two-dimensional.) It took a while to invent two-dimensional mathematics, because we're used to thinking of "functions" acting consecutively in "time", and our experience is that "time" is one-dimensional. Anyway, the point is that if your mathematics is one-dimensional, then it's much easier to see how to break any one-dimensional picture into "basic" subpictures with only two things going on. I think this is the answer to your question 2, why most of the time we only think about 2-ary operations.

Finally, I'll mention that there's another direction you can go, which is to include "coalgebra" along with your algebra. By "algebra" I mean a theory with some "$k$-ary operations" that take in $k$ inputs and spit out one output. But "coalgebra" has operations that have multiple outputs. Coalgebraic operations are very important, especially in computing: you wouldn't want a computer program that only does one thing when you ran it, because then it couldn't also tell you that it had done it!

1. I'd say that a lot of "higher-dimensional mathematics" concerns spaces with operations of arbitrary finite arity. I'm thinking of things like planar algebras, operads, ...
   
   Let me mention one area that I like, which are various things I'd call "associative", and rather than trying to give precise definitions I'll mention planar algebras. A planar algebra includes in the data a $k$-ary operation for every way to draw nonintersecting curves (that are either closed or end on a boundary) on a disk minus $k$ subdisks. These operations are required to compose in any way that you can stick a disk-minus-holes into a hole in another disk (with the requirement that any curves ending on the glued-along boundary components match up). Then there's also an associativity requirement that says that everything only depends on the topology of the diagram, not the geometry.
   
   Anyway, it is possible to write any "planar operation" as a composition of binary operations (although you need infinitely many "basic" binary operations), but this is the wrong way to think about it, I claim. In particular, there's really no canonical choice how to write something as a composition of binaries.

2. From this point of view, let's now revisit usual associative multiplication. The associativity says nothing more nor less than: ab c = a bc. Drawn this way, it's clear that this is again a statement that "only the topology matters, not the geometry". But the point is that the usual multiplication is "one-dimensional", in that the ambiant space where things like "a", "b", "c" are put is a line. (Compare planar algebras, which are inherently two-dimensional.) It took a while to invent two-dimensional mathematics, because we're used to thinking of "functions" acting consecutively in "time", and our experience is that "time" is one-dimensional. Anyway, the point is that if your mathematics is one-dimensional, then it's much easier to see how to break any one-dimensional picture into "basic" subpictures with only two things going on. I think this is the answer to your question 2, why most of the time we only think about 2-ary operations.

Finally, I'll mention that there's another direction you can go, which is to include "coalgebra" along with your algebra. By "algebra" I mean a theory with some "$k$-ary operations" that take in $k$ inputs and spit out one output. But "coalgebra" has operations that have multiple outputs. Coalgebraic operations are very important, especially in computing: you wouldn't want a computer program that only does one thing when you ran it, because then it couldn't also tell you that it had done it!