Does Drush 8.x work with core 7.x?
Yes, Drush 8 works with Drupal 6, 7 and -8.3. I just tested Drush 8.1.17 against a Drupal 7.58 site: worked a charm to clear cache, generate a login URL and download a module.

This is the compatibility table from their site for reference:




  I mean are there different commands to clear the cache for a Drupal 7.x site?


Yes, it was cache-clear, or cc for short in Drupal 7, e.g.

drush cc all