I double-checked, and the media files SHOULD transfer just fine as long as the blog you're exporting is live and functioning as you are doing the import to your self-hosted.
You don't have to delete your current blog, you can just locally host (not connect to the internet, in other words) the exported file - it won't wipe out what you have on your blog when you export!
We call all WordPress.org installations self-hosted, so we use the term freely; however, in your case, if you just keep the installation on your localhost using MAMP and never set up with a server, you don't need to worry about actually hosting your blog yourself (even though you'll be hosting a static version locally).
Just remember, you won't be updating the local version when you update your live WordPress.com version. You can continue to export from your live WP.com blog and import to your local WP.org blog as much as you like, and you shouldn't get duplicate entries.
There may or may not be a plugin for dropbox for WP.org, but we don't currently support third-party plugins for WordPress.com. However, we do have a variety of widgets that are very useful! In this particular instance, however, the thing I think you'll want to do is the export/import.
I hope that helps!