Question: When I am handling the Update Trigger between QBO and SFDC (or any two apps for that matter), do I just map all of the fields that I want to be affected by the update?
The Update Trigger will return all fields for the updated object (for example Invoice) , and if you are creating a corresponding object in another system you have the option of mapping all the fields (you would do this if you creating a new object - for example a new invoice). On the other hand, if you are just looking to update a few fields - for example you want to change the status to paid based on some state change on the other app then you don't have to map all the fields.
If you would like to remove the data in a field in your app, you can use the "clear" formula. Simply toggle the field to formula mode, and type in the word "clear"