I think your best course of action on this would be to create a series of automation rules to handle the different possibilities on requestor types. The automation rule would take care of assigning the ticket out primarily and applying any task templates or workflows. You could then have multiple workflows to take care of the multiple possibilities for a group needing to interact with the ticket. The type of requestor (ie the group the requestor is a part of or their request priority level) would help to determine which automation rule should take precedence and assign the proper workflow while notifying appropriate department heads.
Let me know if this helps.