I want to have my Index page show posts from a certain Category ('news'). I have that functionality, but is there a way I can make a 'News' Page, where the slug just points to the index?
I would take the current code you have and drop it into a page template called tpl-news.php or whatever you choose. Add the following to the header before your index.php code:
Now, create a page called 'News' with a slug of 'news' and select the 'News' page template. That should do the trick for you.
I would also look at how Wordpress template hierarchy works and build template files that target a specific category - Example: category-news.php would force this file to be used when the 'news' category permalink is viewed rather than the default archive template.