Dropdown menus are a part of wordpress.COM as long as you use a theme that has top navigation. They added this capability months ago.
As to the plugins and domain question, no, the domain/domain mapping upgrade just puts your own custom domain name on your blog here. You still have the same restrictions as you did before.
You would have to buy a domain name, hire a web host such as godaddy or bluehost, etc., install and configure the wordpress.ORG software at that web host, and then you could use plugins on that self-hosted blog. Figure a total average monthly cost of $10 to $25 depending on bandwidth and storage requirements.
You will be responsible for all installations and all upgrades, all backups, all troubleshooting. If something goes wrong, you have to figure it out and fix it. If you install a plugin or a theme that kills your blog, you have to know how to go into the backend of things and manually remove it (which sometimes requires that you edit the MySQL database.