Missing "Clean URLs" option
This is a known problem in Drupal 7. The "Run the Clean URL test" button does not behave like that page says it does, and this can be very confusing. Please see "Run the clean URL test" UX is broken; basically, what it comes down to is that the logic and UX of that page is broken, but the clean URL system itself (ie. the part in Drupal core that makes clean URLs work) does not contain any (known) bugs.

If you cannot enable clean URLs (you don't see the checkbox to enable it), this means that a test has already been run, it returned a negative result, and you should check your server configuration. See drupal.org - Clean URLs for documentation.

PS. If you can, test the patch from comment #87. This patch will improve the UX on the clean url's page, but it will not solve problems with an incompatible server configuration.