I downloaded and installed the thing. I read that mods downloaded from the Nexus website would automatically be moved to the right folders. "One-click download, one-click install" is how it's advertised.

I'm not seeing that, yet.

Does the NMM bypass the whole web browser route for downloading? Is there some sort of browser plugin to accompany the NMM?

Right now, it does nothing for me, aside from giving me the load order of my official DLC.