No. WordPress.org and WordPress.com work in totally different ways. With WordPress.com the site is published on the WordPress servers, so you have your username.wordpress.com. You just choose your template and there are limited options for customisation and that is it really. You can pay to add a domain name if you wish, but if you want to do that, then you may as well use WordPress.org.
WordPress.org is what is known as a Content Management System or CMS, this is like Drupal and Joomla. They are open source platforms and are completely free to download and install. If you want to try out WordPress.org you can do this locally on your computer totally free and you don't need to register a domain name or hosting to be able to do this.
CMS's work by using either Wordpress, Joomla or Drupal and a database, where all your website content is stored. To use is locally on your computer, for a PC you need to download WAMP or id you use a Mac, then MAMP - stick with the free MAMP and not MAMP Pro. All you then do is set this up on your computer and then download the latest version of WordPress.org and then place this folder in the htdocs folder under MAMP. You then need to create your database and then connect the two when it comes to installation.
The good thing about WordPress.org is that because you install it on paid hosting, there is more functionality and you can extend and add things, by adding what are known as Plugins - these just extend the functionality of WordPress. The same things are known as Extensions in Joomla and Modules in Drupal.
It is not Wordpress's responsibility to promote your site for you - that is up to you, so no, if you use paid hosting once you have registered your domain name and then the one click install for WordPress that a lot of hosts offer, the rest is up to you.
The way to get your site up there in the rankings is to use interesting content - not just content, but photos and videos too and then SEO, so you optimise your site so that it gets found. Wordpress.org has SEO plugins that you can download and install. Also, register your site with search engines such as Google.