Is a woman's first Hajj considered valid despite her performing its rituals while not in a state of purity, and is it permissible for her to swear an oath that she has not previously performed Hajj if her first Hajj was not valid, in order to perform the obligatory Hajj again?

A woman being in menstruation does not prevent her from entering into ihram for Hajj or 'Umrah.

If she enters into ihram for 'Umrah as a mutamatti' and performs it while menstruating, her 'Umrah is not valid. If she then combines it with Hajj, she has performed qiran (combining Hajj and 'Umrah), and she must offer a sacrifice for qiran. Her Hajj is valid if she performs tawaf al-ifadah in a state of purity and then performs sa'i after it. The farewell tawaf is waived for her if she menstruates.

However, if she enters into ihram as a mufrid (performing Hajj only) or a qarin (combining Hajj and 'Umrah) and performs tawaf al-qudum while menstruating, and then performs sa'i after it, her sa'i is not valid because it was performed after an invalid tawaf. Sa'i is an essential pillar without which Hajj is not valid. She remains in her state of ihram until she performs the major tahallul, and she must refrain from conjugal relations with her husband until she returns to Makkah and performs the sa'i.