Hi,
1. Open JoomSEF's URLs Manager and check the Real URLs of the "Home" SEF URLs. There's probably some variable added by your template or some plugin. Set the variable as "Custom non-SEF variable" in JoomSEF's Configuration or in specific extension parameters.
You can also post some examples so we can check it.
2. Problems with website crawler may have various causes. Does it stop working right after starting, or after crawling some URLs?
3. You can control which URLs are indexed in your sitemap using the Sitemap Manager in JoomSEF. For more information on how to use the Sitemap Manager please refer to JoomSEF's documentation:
www.artio.net/joomsef/documentation#sitemap