Can existing Drupal8 websites be updated using Drush8?

Yes, you can. But use a globally installed Drush 8 version. Simply download the phar file, make it executable and copy it to /usr/local/bin/drush, see http://docs.drush.org/en/8.x/install/

How do I use drush9 on an existing website?

A composer based Drupal 8 installation includes Drush 9 by default and if you run drush inside of such an installation the site-local Drush 9 runs automatically. The global Drush 8 or 9 works in this case only as a launcher.

If you have a non composer based Drupal 8 and you want to run Drush 9 on it, then I would recommend to convert it first to a composer based one.

Can existing Drupal8 websites be updated using Drush8?

Yes, you can. But use a globally installed Drush 8 version. Simply download the phar file, make it executable and copy it to /usr/local/bin/drush, see http://docs.drush.org/en/8.x/install/

How do I use drush9 on an existing website?

A composer based Drupal 8 installation includes Drush 9 by default and when you run drush inside of such an installation the site-local Drush 9 runs automatically. The global Drush 8 or 9 works in this case only as a launcher.

If you have a non composer based Drupal 8 and you want to run Drush 9 on it, then I would recommend to convert it first to a composer based one.