Does JSitemap generate URLs or is responsible for that?
No JSitemap is not a SEF management extension and doesn't generate any URL on yor site.
It aims to generate a mirror of all URLs of your site, based on how these URLs are generated by the SEF system that you are using.
You can choose to use the standard Joomla SEF system or some third party solutions such as sh404sef, JoomSef and others. JSitemap will use exactly the same links available on your website and generated by SEF extensions even for the sitemap. If you experience any issues about SEF URLs format it's more likely that the solution can be found in the SEF engine used instead of the JSitemap generator.