Is the Hadith "Teach your women Surat An-Nur" authentic? And did Sheikh Al-Albani classify it as weak? And what is the authenticity of its other narrations?

The Hadith that commands not to accommodate women in upper chambers, not to teach them writing, but to teach them spinning and Surah An-Nur, is a very weak and fabricated Hadith. This is because one of its narrators is accused of lying, and another's narrations are abandoned.

The other Hadith that commands teaching men Surah Al-Ma'idah and women Surah An-Nur has a weak and mursal chain of narration.

As for the command to teach women Surah An-Nur, it has been narrated from Umar ibn Al-Khattab, may Allah be pleased with him, with a chain of narration whose narrators are trustworthy.