Is it necessary to refrain from eating and drinking after breaking the fast in Ramadan due to a tooth extraction, with continued bleeding on the following day?

Merely extracting teeth is not a sufficient reason to break the fast in Ramadan (definition). It is preferable to do so at night to avoid swallowing blood. If one is compelled to do so during the day due to severe pain, it is permissible, and whatever blood reaches the stomach unintentionally is excused. The questioner must make up the day on which he broke his fast on the pretext of extracting his tooth, while continuing to fast and taking precautions to avoid swallowing blood as much as possible.