Strictly concerning public key exchange methods, do many of these ideas exist? Dozens? Hundreds?
You have to define "exist" more precisely. If you mean widely known and/or standardized primitives, there are the obvious classic cryptosystems such as DH, RSA, ECDH, which would make dozens probably too high of a number to use.
If you mean "exist at all", including the broken ones (as well as the non-broken ones), unpublished ones* by amatuer researchers, and even minor variants of other algorithms, etc, then surely there are at least dozens. Hundreds would be surprising if we are limited to only primitives that facilitate key exchange.
You might consider some algorithms to be variations of others - this could easily influence your count by a large amount. ECDH and DH are an example. In that case, you'd be probably be interested in what different problems and assumptions are used in cryptosystems, and it would probably reduce your count back down to the order of a dozen or so at the most.
Would commercial value exist in creating a new method?
Almost certainly not, at least in the direct and immediate sense. Cryptographic algorithm design is a poor business strategy, because you are competing with widely used, freely available, efficient, standardized algorithms. Unless you come up with something new that nobody else has but everyone wants, patenting your algorithm will probably just ensure that nobody uses it.
You could create specialized hardware for performing your algorithm very quickly or with very little power and sell or license that. But unless your algorithm is standardized and widely used, it is unlikely that it will receive much/any attention.
To this end, is it a problem being pursued in the world today? (Classical computing - I am aware quantum is an academic direction at this time)
NIST's call for Post-Quantum algorithms actually just concluded. So research into it is indeed big at the moment, but if you are were hoping to design an algorithm and have it standardized, you probably just missed your chance.
For the curious, there were 59 KEM/PKE schemes submitted to NIST (according to the previous link).
*Disclaimer: This is my personal github, it just so happens to illustrate my point quite nicely