Is it obligatory to pay a commission to the real estate office after purchasing a house directly from its owner, given that the office was unsuccessful in completing the sale at the price it stipulated, and the purchase was made at a lower price offered directly by the buyer to the owner?

The broker is not entitled to his fee until his work is completed. As long as the real estate office owner has not completed his brokerage work, and the cancellation was not from the client's side, then he is not entitled to anything. If he cancels before commencing the work, or the agent cancels after commencing, then he gets nothing.