It is true that when Lady Hagar searched for water, she was running here and there, hoping to discover something. However, the purpose of the sa’ie is not to commemorate her action in full. Otherwise, what she did would have been clearly marked and we would have been told in plain terms that this is the reason why it is included as a ritual of the pilgrimage and the umrah.
These rituals are done because God wants us to conduct the pilgrimage in this way. When the Prophet (peace be upon him) offered the pilgrimage, he said to his companions: “Learn your rituals from me.” At no point did he say this ritual is a commemoration of this or that act. It is scholars who pointed this out, linking each ritual to an act of a great believer who demonstrated full submission to God.
Moreover, the jogging during the sa’ie is only symbolic. It is for a very short distance, while Hagar was running in the sort of frenzy that characterizes a desperate woman fearing for the life of her young child. She definitely did not walk the rest of the distance, but was running at full speed. We need to demonstrate calmness when we offer worship, and this is what is demonstrated in the sa’ie.
God says in the Qur’an: “God knows best whom to entrust with His message.” (6: 124) We cannot question Him, or ask why this is done in this way and not in some other way. The rule that applies in this regard is: “He cannot be questioned about whatever He does, whereas they shall be questioned.” (21: 23) Otherwise, the questioning would be endless. One could say, for example: since all people are equal, why the last messenger was an Arab, not of some other race? Or why was he sent in that particular generation, not earlier or later? Other questions may be put. It is His message that He wants people to implement and He chose for its delivery the person He knew to be best suited for it.
Having said that, a number of scholars, notably Ibn Hazm, are of the view that the list of prophets included a number of women. They cite evidence in support of their view, but the majority of scholars do not agree with them.
Muslims have always put up some fine calligraphy with verses from the Qur’an, with the intention of reflecting on their meaning and remembering their message. This is perfectly appropriate. Each time we read them we earn a reward. This is different from posting them believing that they bring us benefit or protect us from harm. The fact that one intends to reflect on their meaning and implement their message makes such an action commendable.