What are the cases in which the SLA would not run on tickets?

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The SLA timer will not run on tickets when the ticket is in an SLA OFF status. You could check if the SLA timer for the statuses that you had mentioned is turned OFF under Admin > Workflows > Ticket fields > Status.

Also, the SLA timer would not be running on tickets outside the business hours that you have configured for individual groups. You could check the business hours settings under Admin > Team > Groups.

But the automated emails sent through the email notifications/automation rules would not be considered as responses on the tickets as these are system generated events. Only a reply/public note from an agent would be considered as a first response on the ticket.