Is the manner in which a man leads his wife in prayer the same as when he leads another man in prayer, and what should he do if he forgets to recite aloud in the first raka'ah and recites aloud in the second?

The wife's prayer behind her husband should be with her standing behind him, because it is the prayer of a woman behind a man, and the marital relationship does not change this ruling. The proof for a woman standing behind a man is what Anas ibn Malik, may Allah be pleased with him, said: "So I and the orphan stood behind him, and the old woman stood behind us." There is no difference in this regard between a wife, a mahram, or others. As for forgetting to recite audibly in an audible prayer, it does not affect its validity.