How do I check whether the patch for SA-CORE-2018-002 is applied? How do I check whether the patch for SA-CORE-2018-002 is applied in a Drupal 7 website? 

I have applied the patch. How does it secure my Drupal websites?
As you see in top of the patch, the patch changes the bootstrap.php file, so you can check that file has been changed, and it contains the correct code.

FAQ about SA-CORE-2018-002
 describes why the patch is a security patch.