I'm trying to re-vamp our service desk's ticket templates and want to add a dropdown that lists all of the projects held on our Data Center. I can't seem to find a way to add this field, which is odd to me because it exists on the create issue screen.
Does anyone know if it's possible to add this Projects dropdown to a service desk template or do I need to create a custom field that lists them all?
No. The drop-down you get (for both project and issue type) on the create issue screen is not a field. Project and Issue type are structural attributes of an issue, not fields.
But, there is a custom field type that lets you list and select from your project list. Go to admin -> Issues -> add new field, select "all" or "advanced" and start typing "project" into the search box. You'll find a project-picker.