Also “Join node to way” and “Move node onto way” now operate on multiple nodes (I coded the enhancement myself).

I’ve found that it works wonders to clean certain type of geometry: Just select a bunch of nodes that should be part of nearby ways, adjust the zoom so that such nodes are 10px or less from the ways you want them to be part of, and execute “Move node onto way” (works best if you assign a shortcut for it). It can be restricted to snap only to certain ways by selecting those ways as well.