Is the aforementioned Hadith regarding the gates of Hell and the authenticity of the story of the Bedouin woman sound, and have other Hadiths been reported concerning the gates of Hell?

Yes, Hell has seven gates, as stated in the Holy Quran: "And indeed, Hell is the promised destination for them all. It has seven gates; for every gate is a [specific] portion [of them] determined." (Surah Al-Hijr, 15:43-44). And in the Hadith: "Paradise has eight gates and Hell has seven gates." Al-Bayhaqi mentioned their names: Jahannam, Lazha, Al-Hutamah, As-Sa'ir, Saqar, Al-Jahim, Al-Hawiyah. As for the hadith mentioned by the questioner regarding the liberation of a nation, it is weak and not authentic. The correct account is that a woman heard the verse and freed seven slaves for the sake of Allah.