In the settings panel there is a button to rebuild the index. What happens when you do that? If we have saved custom stuff in the index (for example when saving a node), do we lose that information? If so, is there any way to trigger such custom indexing stuff outside of for example a ContentPublishedNotification?

Well it basically rebuilds the index that you've configured, you can add custom properties to your index if needed. But if its just a regular document type property that gets updated on a notification, the newly build index should contain the new value too (the button basically collects all documenttype and ipublishedcontent data and indexes it, simply said)