One Identity 8.1 Template Triggers - API

Normally not. But it depends on how you update the Person record and how your templates are defined. 

What about updates from a SyncProject? They also are not firing the templates. It must be related to the templates themselves. What needs to be changed/updated in my "LastName" template for example (set to overwites) that might prevent them firing?: 

Dim Last As string
     Last = CCC_GetPreferredLastName($PreferredName$)
If Last = string.Empty Then
     Last = $LastName$
End If
Value = Last

What about updates from a SyncProject? They also are not firing the templates. It must be related to the templates themselves. What needs to be changed/updated in my "LastName" template for example (set to overwites) that might prevent them firing?: 

Dim Last As string
     Last = CCC_GetPreferredLastName($PreferredName$)
If Last = string.Empty Then
     Last = $LastName$
End If
Value = Last

If this template placed on LastName column, then you need to turn on Overwrites otherwise the template wouldn't touch the value if a value already exists.

In regards to the Synchronization, I refer to other threads, the documentation and a knowledge article from support.

https://www.quest.com/community/one-identity/identity-manager/f/identity-manager-forum/27254/synchronization-and-templates

https://support.oneidentity.com/technical-documents/identity-manager/8.1/target-system-synchronization-reference-guide/7#TOPIC-1130093

https://support.oneidentity.com/de-de/identity-manager/kb/227832/template-is-not-executed-on-a-data-import-via-a-synchronization-

Yep - checking Data Import did the trick. Wondering if what this check box does exactly is documented? 

You might want to check the links in my answer. One of them is pointing to the documentation of the data import flag.