If I understand correctly you have main web site which is not WordPress, but you want to implement WordPress Blog inside it, and create premalinks/categories as you described.
If I were handling this requirement, I would go about it by installing separated WP instance on subdomain, such as blog.example.com , and then create categories like this
blog.example.com/category/news/
blog.example.com/category/career/
blog.example.com/category/case-studies/
Additionally, you can create HTTP 301 redirection(s), and “map” some virtual URL-s to your blog. For example, you can map www.example.com/category/news/ to blog.example.com/category/news/ , so if user visits category on main page it will be redirected to appropriate sub domain.
Technical details:
HTTP Redirection(s) are usually created inside .htaccess file, good tutorial about this can be found on page, http://www.inmotionhosting.com/support/website/redirects/setting-up-a-301-permanent-redirect-via-htaccess
Example HTTP 301 redirection would look like this:
Redirect 301 /category/news/ http://blog.example.com/category/news
Important note: Creating multiple WordPress installations just to cover different categories is not an option. It will cause you various of issues in future (updates, same design, multiple plugins, analytics, tracking).