You can see whats available by standard in https://confluence.atlassian.com/servicedeskserver033/configuring-service-desk-notifications-861255295.html
There is some feature requests.
https://jira.atlassian.com/browse/JSD-4642
To meet all your requirements its most likely a plugin needed.
Few examples,
https://thepluginpeople.atlassian.net/wiki/display/JEMH/Integrate+JEMH+with+JIRA+Service+Desk
https://marketplace.atlassian.com/plugins/com.riadalabs.jira.plugins.notificationassistant/server/overview
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