I want to be able to access my Office Pool from two different computers and I will be using DropBox. Do I need to put the templates and other data found in Library: Application Support, onto each computer? If so, do is there an way to sync them automatically?

Unfortunately, each computer will have to have the templates in Application Support. And no, OfficePool will not automatically find these so you will need to sync theme manually. A future version may have these templates embedded within each file but that is not the case at this time.