Technically speaking webpart properties are NOT indexed. The content from that webpart field are stored on the document level and not at the webpart. So it is the document's content that is indexed. This is why I rarely use editable text, static text, etc. webparts. You have little to no control over what is indexed with them. Use a page type for this type of scenario.
To your example of the accordion webpart, again, these fields (accordion panes) are storing the content in the document not in the webpart. Take a look at the actual database table. You'll find in the 'cms_document.documentcontent' field, there is a reference of the webpart id and the value which was entered. Again, even though it looks like the webpart is storing the content it is not; the document stores the content and uses the webpart ID to populate the webpart when used for editing and also displaying properly on the live site.
Also if you're doing a Page Crawler search index those are much harder to filter out the content you are searching because it searches all the rendered content whether or not the content is hidden or not.