Q 1: What is
the ruling of Islam on answering the greeting of Christians, following their funerals, and offering them condolences?
If a Kafir (disbeliever) greets a Muslim, the latter should reply by saying "The same to you" as mentioned in the Sahih (sound) Hadith in which the Prophet (peace be upon him) said,
When the People of the Book greet you, you should say, "The same to you."
However, it is not permissible for a Muslim to follow the funeral of a Kafir, for this is considered an act of loyalty to them which is Haram (prohibited). However, consoling them is acceptable if there is a Shar`y (Islamically lawful) probable interest. In this case, they should say, "I offer you my condolences, may your distress be relieved," but they should not say, "May Allah forgive your deceased," for seeking forgiveness for a Mushrik (one associating others in worship with Allah) is not permissible.