Is the Hadith, "Whom among women do you love most? He said, Aisha. And whom among men do you love most? He said, her father," used as evidence for the permissibility of a man mentioning his wife's name in front of his friends?

There is no religious impediment to mentioning one's wife's name in front of men. This is supported by the agreed-upon hadith (definition) of Amr ibn al-'As, where he asked the Prophet, peace and blessings be upon him, "Who is the most beloved person to you?" He replied, "Aisha." Then he asked, "And then, among men?" He said, "Her father," and then mentioned other men.