Thanks for putting up your question on Just Answer.
Without actually examining you or at last seeing a good quality close up photo of the extraction area, I cannot comment what the piece of bone is. Based on your symptoms though it looks like you do have Dry Socket.
A dry socket is typically the result of something that has dislodged the normal blood clot, such as smoking, drinking through a straw, brushing the area, or trying to clean the extraction site. It isnt really dry, but given the name as the extraction socket doesnt have the blood clot and granulating (healing) tissue. I can imagine the excruciating pain you must be experiencing. The extraction socket takes a long time to heal , with a gaping hole is seen for a few weeks. The pain should subside after 10-12 days following the second surgical date (6 September).
Im sorry, but for Dry Sockets, patience is the key. Avoid hot, spicy and hard foodstuff. Take more of cold, sweet and soft food.
A zinc-oxide eugenol based dressing placed in your extraction socket has been shown to help. This would have to be reviewed periodically and changed if required.
Hope this answers your question. Please leave a positive rating for me if this answer has helped you. I do not get credit for it otherwise.
Good Luck !