Praise be to Allah.
This question covers a number of issues
When you should not read the Quran?
It is permissible for a Muslim to read Quran in all situations, except when he is junub (in a state of major ritual impurity following sexual activity). Shaykh al-Islam [Ibn Taymiyah] said: “The scholars unanimously agree that it is forbidden for the person who is junub to read the Quran.” (See Tawdih al-Ahkam by ‘Abd-Allah al-Bassam, vol. 1, p. 309)
With regard to what is mentioned about reading the Quran being dhikr, this is correct, indeed it is one of the best forms of dhikr, because it is the words of Allah.
Forbidden time to offer prayer
The Prophet (peace and blessings of Allah be upon him) forbade prayer at certain times . It was narrated that it is not allowed to pray or to bury the dead at those times , but that does not include reading the Quran. Rather it was narrated from the Prophet (peace and blessings of Allah be upon him) that it is mustahabb to remember Allah (dhikr) after Fajr, which undoubtedly includes reading the Quran.
Al-Tirmidhi narrated that the Prophet (peace and blessings of Allah be upon him) said: “Whoever prays Fajr in congregation, then sits and remembers Allah until the sun comes up, then prays two rak’ahs, will have a reward like that of Hajj and ‘Umrah .” He said: the Messenger of Allah (peace and blessings of Allah be upon him) said, “Complete, complete, complete.” (al-Jumu’ah 535; classed as good by al-Albani in Sahih Sunan al-Tirmidhi, 480).
Disallowing this is not valid unless there is evidence to that effect; how can that be when there is evidence to indicate that it is mustahabb?
Is sujud al-tilawah considered to be prayer?
Shaykh Ibn ‘Uthaymin said:
Some of the scholars were of the view that sujud al-tilawah is prayer. Some of the scholars were of the view that it is not prayer, because the definition of prayer does not apply to it, as it is not proven in the Sunnah that there is any takbir (saying “Allahu akbar” as in the beginning of prayer) or taslim (saying salams, as at the end of prayer) involved in it.
The hadiths narrated concerning Sujud al-Tilawah do not mention anything except sujud (prostration) – “he prostrated and we prostrated with him” – except for one hadith which was narrated by Abu Dawud, but its isnad is subject to question (“he said ‘Allahu akbar’ when he prostrated”). But there is no taslim.
It was not narrated in any hadith, da’if (weak) or sahih (sound), that he said taslim after doing sujud al-tilawah. If there is no sahih hadith to indicate that one should say taslim, then it is not prayer, because prayer has to begin with takbir and end with taslim.
This is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him). On this basis there is no need for taharah (to have wudu) or to cover one’s ‘awrah or to face the qiblah.
It is permissible to prostrate even if one has broken one’s wudu (minor impurity). Whoever studies the words of Ibn Taymiyah will see that the correct view is that which he expressed, which is that sujud al-tilawah is not prayer and is not subject to the same conditions as prayer.
If you recite the Quran from memory and you do not have wudu , and you come to a verse where sujud is required, then according to this view you may prostrate and it is OK. Ibn ‘Umar (may Allah be pleased with him), even though he was known to be very strict, used to prostrate when he did not have wudu, but to be on the safe side it is better not to prostrate without having wudu. (Al-Sharh al-Mumti’ by Ibn ‘Uthaymin, vol. 4, p. 126)
On this basis, it is permissible to read the Quran after Fajr prayer, indeed it is mustahabb.
With regard to the issue of the reader coming across a verse where sujud al-tilawah is required, after Fajr or after ‘Asr, Shaykh Ibn Baz (may Allah have mercy on him) was asked whether the person who is reading the Quran after ‘Asr or after Fajr should do sujud al-tilawah. He said:
“It is prescribed to do sujud al-tilawah after Fajr and after ‘Asr, because this is something that is done for a reason, and because it does not come under the rulings of prayer according to the more correct of the two scholarly views. Rather it comes under the ruling of reading Quran and saying tasbih (“subhan Allah”) and tahlil (“la ilaha ill-Allah”), so it is permissible for the one who is reading Quran to do sujud al-tilawah, even if he does not have wudu, according to the more correct of the two scholarly opinions.” (Fatawa al-Shaykh Ibn Baz (may Allah have mercy on him), vol. 2, p. 344)
And Allah knows best.