Is water into which a hand was dipped after touching soap —which had fallen into the bucket of water— considered impure? And is it necessary to repeat the ghusl if the wife was menstruating and did not inform [her husband] until after completing the ghusl?

The fundamental principle is that water is pure and purifying, and it only becomes impure if its color, taste, or smell changes due to an impurity. A menstruating woman dipping her hand into water does not affect it or make it impure, because a Muslim's body is pure, whether in a state of janabah (definition) (major ritual impurity) or menstruation, based on the Prophet's (peace and blessings be upon him) saying: "Indeed, a believer does not become impure." Therefore, the questioner's ablution is valid, and she is not required to repeat it.