Is it true that Abu Bakra was flogged by Umar for his testimony against Al-Mughira ibn Shu'ba on charges of adultery, and does that affect the authenticity of the hadith "Never will a people prosper who entrust their affairs to a woman," which was narrated by Abu Bakra?

Abu Bakra was a venerable Companion with numerous narrations. Al-Mughira was accused of adultery, and Abu Bakra, along with a group of other Companions, testified against him. Although Umar subjected them to the hadd punishment for slander (qadhf) due to the insufficiency of the testimony, this did not invalidate Abu Bakra's uprightness (ʿadala) or his narrations.

This is because the hadd punishment for slander is of two types: The first is due to outright falsehood, which invalidates one's uprightness. The second, however, is due to the insufficiency of the testimony, not falsehood, and this does not invalidate one's uprightness. Abu Bakra's case falls under the second type.

None of the Salaf (early generations) impugned Abu Bakra's uprightness or his narrations; rather, scholars unanimously accepted them.

Furthermore, all the Companions are considered upright (ʿudūl). Allah and His Messenger have vouched for them, and they do not require validation from anyone else. Certainty is not superseded by doubt.