Do we need to update every release sequentially to get to the latest version? Drupal releases a minor release approximately every 6 months, and a patch release every 1 month. Do we need to update to each one of them sequentially in order to get to the latest version? E.g, if I'm on Drupal 8.5.0, and the current Drupal version is 8.6.1, do I need to follow the order and update each release in the order 8.5.0 -> 8.5.1 -> 8.5.2 -> 8.5.3 -> 8.5.4 -> 8.5.6 -> 8.6.0 -> 8.6.1, or can I directly update to 8.6.1 from 8.5.0 by replacing the core files?

If we can update to 8.6.1 directly, how do we handle the case if a release like 8.5.4 requires changes to 'default.settings.php' but 8.6.1 doesn't require?
You do not need to update Drupal sequentially. The Drupal update system will run all the needed updates so that you can go from (lets say) 8.3.0 to 8.6.1 directly. 

Regarding your question about "If we can update to 8.6.1 directly, how do we handle the case if a release like 8.5.4 requires changes to 'default.settings.php' but 8.6.1 doesn't require?" ... If 8.5.4 requires a change to settings.php then 8.6.1 will assume that those changes are there already. 

Note: its a good practice to update core (at least for minor versions) so that you are always in a position to quickly apply security releases. The Drupal security team has recently decided to support the most recent 2 minor releases, but still...

Check out this further reading and also this