Your theme will not transfer over. The themes here have been modified to work with the unique features and restrictions here at wordpress.COM, so they typically will not work on a self-hosted blog. Most all themes used here though are also available for self-hosted blogs. You can start with a search for your theme at http://wordpress.ORG/extend/themes/ . If it is not listed there, then do a general internet search somewhere like google with something like this in the search box "theme name wordpress theme."
Since you have domain mapped your domain name here, when you switch over, all internal links will transfer over just fine. During the import process from the self-hosted blog, you will be given the opportunity to have it pull all of your images over to the new installation. All image URLs should be corrected during the import process. Now the disclaimer: The above does not always work flawlessly and in some cases you might have to manually fix some links. This seems to be more of an issue lately than it has in the past.
When you do the export from here and import it into the new blog, all of your data will still also be here. What you will want to do is set the blog here to private so that the search engines do not see any duplicate content.
There really isn't anyway to have multiple "blog" pages. There is one main blog page, which will contain all of your posts in all categories. There are ways to accomplish this though. You use categories for all your different subjects or divisions and then put the categories into the main navigation for your site. Then when someone clicks on the category name, they will see only posts on that subject. If you do not want the typical "catch-all" blog page, you can set a static front page for your blog with a welcome message or whatever, and the you visitors can navigate to the categories via the main navigation.
One note: There are a lot of unique widgets here that you will not find in the base self-hosted installation. Some of that functionality can be recovered with the new "jetpack boost" thing, which will bring back some of the features. The rest, if you need them, will have to be accomplished with plugins.