I would like to know how I can add a JavaScript action that would act on all search results (like the "Save New Search" in the search results' "More Actions") and how the search results can be passed as a parameter to that action. I didn't find anything in the "web-client-config-actions.xml" configuration file.
That's not currently possible via a configured action. The only way to do something similar would be to code the same Lucene search into the javascript directly or pass it on a URL to a REST style JavaScript via the command processor servlet.