In the Make Offer plugin does it allow to show the price and a sold marking or flag if the item is marked as sold ?

The idea would be to allow best offer and if or when business is made, publish the price for with the trade was done and to show that it was sold.

No.
Accepting offer does not mean item is sold.
User may be selling multiple same item or just have more quantity.

Many themes has feature to mark listing as sold.