Is a woman's marriage valid if she marries another man during her 'iddah, and the second man has not consummated the marriage with her, knowing that she had one menstrual cycle during the 'iddah?

Firstly: Invalid marriage (نكاح باطل) is what scholars unanimously agree on its incorrectness, such as marriage to maharim (unmarriageable kin). Corrupt marriage (نكاح فاسد) is what they differed on its correctness, such as marriage without a guardian (wali).

Secondly: If the first marriage contract was concluded and you both believed in its validity, then you separated by divorce, you must observe the full waiting period ('iddah), which is three menstrual cycles.

Thirdly: If you both believed the marriage to be unlawful, or you were ignorant of the ruling, then the unlawfulness became clear to you, and separation occurred because of it, scholars differed regarding the waiting period. The madhhab of the Shafi'is and Hanafis is that she does not have to observe an 'iddah. The madhhab of the Malikis and some Hanbalis is that she must observe the 'iddah like a divorced woman. Sheikh al-Islam Ibn Taymiyyah and Ibn 'Uthaymeen chose the waiting period of istibra' (ensuring the womb is empty) by one menstrual cycle. In this case, if a marriage contract was concluded for her before the end of her 'iddah, it is not valid, because it is not permissible to marry a woman observing 'iddah, whatever her 'iddah may be.