Yes the sitemap is updated automatically when new contents, menus, etc are added to your components, and search engines will be able to fetch updated informations automatically. The only exception is that you are using a cache system, in such case obviously you need to refresh the cache.
The static sitemap.xml file is no more necessary using JSitemap. This solution is now deprecated, because too complex to manage, having to generate a new sitemap and refresh the file manually everytime your sitemap changes. This is used only in particular cases, such as splitted sitemaps.
To overcome this limitations JSitemap introduces the new concept of dynamic generated sitemaps, using direct links to the component to produce an XML sitemap on the fly, being it always updated automatically.
To ensure that the sitemap will be found by all search engines, it's enough add the entry for the XML sitemap to the robots.txt file, this can be done in just one click using the 'pen icon' button of JSitemap in the main SEO dashboard.