The purpose of our creation is worship, Allah Almighty says:
And I have not created the Jinn and the men but that they may worship Me.(Surah al-Dhariyat, 51:57)
During menstruation, women are given a break from the formal Salat because of its specific requirements, but no one is given a break from fulfilling the purpose of human creation, whether man or woman. Allah Almighty describes believers as:
Those who remember Allah while standing, sitting, and lying on their sides.(Surah Al-e-‘Imran, 3:192)
O ye who believe! Remember Allah with much remembrance; And glorify Him morning and evening.(Surah al-Ahzab, 33:42–43)
O ye who believe! Let not your wealth and your children divert you from the remembrance of Allah.(Surah al-Munafiqun, 63:10).
The Holy Prophetsas said:
The example of the one who celebrates the Praises of his Lord in comparison to the one who does not celebrate the Praises of his Lord, is that of a living creature compared to a dead one.(Sahih al-Bukhari, Book 80, Hadith 102)
Worship through Dhikr-i-Ilahi (remembrance of Allah) is a daily and uninterrupted feature of every believer’s life, whether she is menstruating or not. In the story of Hadrat Maryamas, Allah Almighty commanded her to particularly engage in the remembrance of Allah immediately after childbirth, saying:
So eat and drink, and cool thy eye. And if thou seest any man, say, ‘I have vowed a fast to the Gracious God; I will therefore not speak this day to any human being.(Surah Maryam, 19:27)
In commentary, Hadrat Musleh Mau‘udra wrote:
From this, we also learn that remembrance of Allah is not for- bidden during postpartum bleeding and menstruation. People mistakenly think that a person cannot even observe remembrance of Allah in their heart in such a state, although if remembrance of Allah is forbidden, then spirituality completely dies.(Tafsir-e-Kabir, vol. 5, p. 187)
The offering of durud also applies to every believer at all times, man or woman. We observe durud every time the name of the Holy Prophetsas is mentioned. The Holy Prophetsas said:
A miser is one who does not invoke blessings upon me when I am mentioned in his presence.(Sunan at-Tirmadhi, vol. 6, Book 45, Hadith 3,546)
Durud is an essential part of all of our formal and informal prayers. Hadrat ‘Umarra said:
Indeed the supplication stops between the heavens and the earth. Nothing of it is raised up until you invoke blessings upon your Prophet.(Sunan at-Tirmadhi, vol. 1, Book 3, Hadith 486; Conditions of Bai’at, p. 62)
Durud is a daily and uninterrupted feature of every believer’s life, whether she is menstruating or not. The elder daughter of the Promised Messiahas narrates about the wifera of the Promised Messiahas:
I remember from my childhood that during the days in which she would not be observing Salat, she would not waste that Salat time in conversation. With regularity during the set times of Salat, she would pray and observe remembrance of Allah while pacing and sitting. I always saw her practicing this.(Sirat o Swanih Hadrat Amman Jan, p. 688)
As for recitation of the Holy Quran while menstruating, once Hadrat Amman Janra observed a woman, who was menstruating at the time, returning from Dars-ul-Quran [Quran Study] while carrying a copy of the Holy Quran. She asked the Promised Messiahas if it was permissible to touch the Holy Quran in this state. Heas said:
When Allah Almighty has given leave during these days, then why shouldn’t we? Tell her not to read the Holy Quran during these days.(Siratul-Mahdi, vol. 2, p. 243)
Whilst being in her menstrual period, a woman may recite the memorized portion of the Holy Quran in her heart as a form of remembrance of Allah and if necessary she may hold a copy of the Holy Quran in a clean cloth. Likewise, she may read some portion of the Holy Quran if a particular reference needs to be given to someone or if she requires to teach it to children, but she is not allowed to formally recite the Holy Quran.
Similarly, she is not permitted to recite the Holy Quran on a computer screen even though she may not be holding it directly. However, she may use this facility for instance to find references from the Holy Quran or show certain references to others if needed. There is no harm in doing so.