I have read that it's important for a search engine to have access to your sitemap for better indexation. How can I submit this sitemap and more importantly, how can I actually generate an XML sitemap?
I do not see any necessities for submitting your site map to every search engines, I think you should focus only on a few like Google or Bing. For Google, go to Google Search console, create an account and add your property. For Bing go to Bing Webmaster Tools, create account and submit your XML site url to the sitemap section.
Click on SITEMAPS and submit your sitemap URL. Usually it's either website.com/sitemap.xml, website.com/sitemap_index.xml. If you enter website.com/sitemap.xml in your browser, the SEO plugin should take you to the right page. Here is how the GSC interface looks for this:
Site map are submitted for improving search engine indexing. Sitemaps will help search engine algorithm to learn about your website. But whether you have submitted your site map or not, your website will be still indexed in search engines (unless you have disabled this option).