I think that if everyone in your army has armor, you're not a knight, you're a soldier.
I think maybe the Nazgul could be Anti-Paladins. Boromir and Faramir are the closest things to knights, even if Faramir is also the head of the Southern Rangers. (Faramir is kind of a Ranger/Cavelier multi-class.)
I think the elves are fairly magical - not that all of them are magic-users, but that some of them are, certainly all of the leaders and elders.
I'd never really thought about knight's or the fool's errand in regards to Tolkein before, but it seems that Bilbo/Frodo is the corollary (not perfectly) of Parcival. Gandalf and Merlin are easy to tag. There is no direct corollary between Arthur, Lancelot or Guenevere - I would say that Aragorn is more like Sir Gawain.
Hmmm, someone must have written about this, considering how famously Tolkein talked about creating a new myth for the ages. I guess I have some research to do and blog posts to write.