Is it permissible to pray, fast, perform Hajj, give charity, or supplicate for mercy for my father whom I have never seen pray or fast throughout my life? And is a childhood friend who was treacherously killed considered a martyr if he was regular in prayer, or is that not a condition?

It is permissible to seek forgiveness for your father who is negligent in prayer, charity, Hajj, and fasting on his behalf, and the reward for that will reach him. However, it is not permissible to seek forgiveness for someone who abandons prayer out of denial of its obligation. Prayer is not performed on his behalf with the intention of making up for missed prayers, but it is permissible to dedicate the reward of righteous deeds to him. One who is wrongfully killed is considered a martyr, but the majority of scholars hold that such a person is washed and prayed over, unlike the martyr of the battlefield.