Is it really necessary to include a root (homedir) of the site to the sitemap? As I can understand the sitemaps main idea - it is not any necessary. Am I right?
It is not necessary since it is going to be indexed anyway. Although, It is always good idea to be thorough. And including one more link/page on the sitemap is not difficult.
Remember, the idea of the sitemap is to tell the crawler that you want those pages crawled, plus some other information to help the crawler save some work and also keep your site fresh on the index. That doesn't mean that it's going to index all of them or that only those are the ones tha are going to be indexed, there are algorithms working there that will try to index every indexable page on the site.