I say, with Allah's guidance: It is not permissible to pray for mercy and forgiveness for a non-Muslim after death, because they are subject to punishment, not grace, due to their abandonment of faith in Allah Almighty. Mercy is only permissible for a Muslim, as they are eligible for Allah's mercy, while a disbeliever, due to their disbelief, is subject to Allah's wrath. Allah Almighty said: {And never pray [the funeral prayer] for any of them who has died, nor stand at his grave. Indeed, they disbelieved in Allah and His Messenger and died while they were defiantly disobedient}[At-Tawbah:84].
In Al-Muhit Al-Burhani 2: 184: "Seeking forgiveness for a disbeliever is forbidden. Allah Almighty said: {Ask forgiveness for them or do not ask forgiveness for them. If you should ask forgiveness for them seventy times - never will Allah forgive them. That is because they disbelieved in Allah and His Messenger, and Allah does not guide the defiantly disobedient people}[At-Tawbah: 80]". Mercy does not differ from seeking forgiveness, as their mercy after death is through the forgiveness of their sins.
As for during life, one may pray for their guidance, as mentioned in the letter of the Messenger of Allah  to Heraclius: (In the name of Allah, the Most Gracious, the Most Merciful. From Muhammad, the servant of Allah and His Messenger, to Heraclius, the great of the Romans, peace be upon those who follow the guidance...) in Sahih Bukhari 5: 2310,  and the Prophet  said: "O Allah, guide Daws and bring them" in Sahih Bukhari 4: 44, and in Tuhfat Al-Muluk 3: 97: "If one says to a non-Muslim: May Allah prolong your life, it is not permissible, unless he intends the prolongation of his life for his Islam." 
This does not contradict wishing well for non-Muslims, and the greatest good is their guidance to the truth, as we dislike their disbelief which harms them and humanity, and contradicts the good for people and includes their harm, and Allah knows best.