Is the use of a kunyah for men Sunnah, مستحب (recommended), or a matter of appreciation? And why is mentioning a woman's name considered shameful in some countries, even though the Qur’an mentions prophets and women by their names?

Using kunyas (honorific nicknames) is an ancient Arab custom meant to show respect and esteem. The ruling on this matter depends on custom (urf). A person should be addressed by their kunya if they prefer it. As for mentioning women by their names, there is no harm in it. However, custom should be observed if doing so is considered shameful among unrelated men.