To have your blog page be example.com/news, create a Page in the Dashboard called "News". Then under reading settings, set the posts page to static and select "News" from the drop-down list.
That leads us into...
1. Turn "front-page.php" into a page template. (http://www.wordpressmax.com/customize-wordpress/custom-page).
2. Create a new Page in the Dashboard and set it to use that template you created in step one.
3. Go to Reading Settings and set that page you created in Step Two as your home page.
Yes, you still need index.php. index.php and style.css are the two files that all WordPress themes must have. And since you're using custom templates, you must have an index.php file.