If I set a task with the setting 'Do not send the user a new task notification email', does this stop all notifications for this task?

The option Do not send the user a new task notification only applies to the “new task” email (the email that is sent right away when the task gets issued).

This setting does not stop reminder emails from going out to those who have been tasked with content.

So the relevant users would still receive the following reminder emails depending on what options are enabled/disabled in the Advanced Settings when creating the task or edited after the task is created:

Due in 7 days

Due in 3 days

Due tomorrow

Overdue.