RE: Finishing Production Order in not possible without defining inventory posting group with blank location
On the Production Order, in the Posting Fast Tab, there is a field "Location Code". You need to fill this in. No need to change your Inv. Posting Setup.
This determines where the Output of the Production Order is going to land. It sounds like you have a Location established in your company (which is correct), but you have not assigned it to the Production Order. This is often the case with manually created Production Orders, as they don't have a place to pull the Location Code from.
Using the Planning Worksheet or directly from the Sales Order are other ways to create Production Orders, which will fill the Location Code based on where the demand is.
I would recommend you also turn on Location Mandatory in your Inventory Setup to avoid posting to or from a blank Location. Finally, I would recommend against setting up an Inventory Posting Setup with a Blank Location, this will also avoid posting entries to a blank Location.
Build into your workflow process documentation that the Location Code on the Production Order should be filled in prior to Refreshing the Production Order.