Categories are meta data for n:n relations: Each category can be related to multiple posts, and each post to multiple categories. So categories aren’t what you need.
I would set up a multi site installation. Use the main site for everything related to all sub sites and the subdomains or sub blogs for each section.
To force the same theme to all blogs, just delete the default theme (TwentyEleven, in 3.4 probably TwentyTwelve), install your custom theme and add the following line to the wp-config.php:
define( 'WP_DEFAULT_THEME', 'folder-name-of-your-custom-theme' );
Each new blog will now use this theme.