What must I do, and are the first and second Umrahs valid, given that I entered into a state of ihram, performed the sa'i, and exited ihram while menstruating during the first Umrah, and then entered into a state of ihram from Al-Taneem and performed the second Umrah after purification on the day of travel?

The questioner sister was obligated to perform Tawaf, Sa'i, and shorten her hair after her purity to exit the state of ihram from her first Umrah. Her entering into ihram for a second Umrah while she had not completed the first is invalid. Thus, the second Umrah is void, and her Tawaf and Sa'i are considered a completion of the first Umrah. Scholars have explicitly stated that it is impermissible to perform one Umrah immediately after another. Her first Sa'i is invalid because it occurred before a valid Tawaf. Her Umrah was completed with her second Tawaf and Sa'i and the shortening of her hair. If she committed any of the prohibitions of ihram before completing her Umrah, then it is more cautious to offer a fidyah (expiation).