You can enable/disable email notifications by navigating to Admin-->General Settings-->Email Notifications. However, please note that there are few system notifications which can’t be disabled.

You would need to add agents to the notify agents section available under Admin-->General Settings-->Email Notifications-->New Ticket Created(Agent Notification).

Click on the edit button next to notify agents. Here you can choose the particular agents who need to be notified when a new ticket has been created.
We would not be able to enable ticket creation email notifications for non-agents.
If an agent is not receiving notifications for new ticket creation, please check with the administrator if the agent’s name has been added to the ‘Notify’ list under Admin-->General Settings-->Email Notifications-->New Ticket Created (Agent Notification)
You can disable the “User Activation Email” notification under Admin-->General Settings→ Email Notifications.

Automated notifications that get triggered from your account are available under Admin-->General Settings→ Email Notifications. You can click on the respective notification and can modify the contents to suit your requirements.
There are default notifications available for requesters under Email Notifications which notify the requester when the status of a ticket is updated to either Resolved or Closed.
We would not be able to stop the close ticket notification for a particular ticket. However, if you use our keyboard shortcuts as below, the closure notification would be skipped for that particular ticket. Try closing the ticket by using the Keyboard shortcut (Alt + ~) or you can press Shift and click on the close button on top of the ticket

We have both First Response SLA notification and Resolution time SLA notification available under Email notifications which can be edited to include custom information if needed. You can navigate to Admin-->Email Notifications and find the notification templates under Agent Notification.

Please note that these notifications work only when you have configured Ticket Escalations setup in your SLA policy
We have a default notification available for requesters under Email Notifications which notifies the requester when an agent adds a public note to the ticket.