Chromium doesn't seem to have any obvious MIME-type configuration settings, which strikes me as a little broken. My specific problem is that there are a number of filetypes that Chromium plays in-browser that I'd rather either download or associate with whatever my system-default MIME-type would suggest. Can this be done?
Look at this ArchWiki entry in the section on File associations. To quote from there:
Unlike Firefox, Chromium does not maintain its own database of mimetype-to-application associations. Instead, it relies on Xdg-open to open files and other mime types, for example, magnet links. See Xdg-open for more information.
There are exceptions to this rule though. In the case of mailto URIs chromium calls out to xdg-email which is similar to xdg-open. Other protocol handlers may have equivalent scripts so check /usr/bin/xdg*. ...