1. Is it permissible to cut ties with a brother who does not pray or fast, despite his good manners, or should the relationship with him be limited to certain boundaries? 2. What are the limits of interaction with a father who does not pray or fast but is good-natured and helps people, and what should the questioner do when the father quotes the verse: "Say, 'O My servants who have transgressed against themselves...'"? 3. Is it permissible to delay Fajr prayer after a late awakening in order to do something that occupies the mind and affects devotion? 4. Is the questioner considered a severer of kinship ties with his ill-tempered and troublesome sister if his interaction with her is limited to a daily greeting of peace to avoid disputes?

You must continue to advise your brother and your father who are committing the sin of abandoning prayer and fasting, reminding them of the gravity of this and Allah's punishment, and clarifying that abandoning prayer is more severe than other major sins, and that its abandoner is considered outside the fold of Islam by some scholars.

Regarding your brother, boycotting him or maintaining ties with him depends on the Sharia interest. If boycotting leads to his repentance, then it is preferable. If it increases his persistence in sin, then maintaining ties is preferable.

As for your father, his right is great, and you must honor and be good to him, for your kindness may be a cause for his guidance.

Your father's argument using the noble verse is incorrect, because the verse pertains to those who repent. As for persisting in sin and hoping for forgiveness, that is a sign of abandonment (by Allah).

The time for a prayer that a Muslim sleeps through is upon waking up, and it is safer to make it up immediately to avoid dispute, unless there is an excuse such as relieving oneself of the two impurities or the presence of food.

There is no harm in abandoning anything beyond a greeting in maintaining ties with your sister if it results in harm; otherwise, it is obligatory to maintain ties with her in the customary manner.