I see many Muslims who do not apply all of it, and see no problem with it. For instance, plenty of Muslims have non-Muslim friends, while the Quran advises not to. More obvious examples of this are homosexuals who claim to be Muslims. Hence my question: do these people have a school? I looked for one and I could not find any. I am not asking about the particular cases I mentioned, but rather for a school which does not think the entire Quran is the literal word of God.
Summary: Muslims universally believe the Qur'an is literally the word of God, and this is part of the third article of faith, "Belief in His Books" (IslamWeb). However, this is distinct from insisting that Allah is restricted to only speaking literally in the Qur'an, and indeed Qur'an 3:7 warns against a too-literal interpretation. Finally, there are ayat of the Qur'an which are not taken literally.
Reliance of the Traveller a4.3 gives examples of scholars not taking literally attributes of Allah described in the Qur'an:
a4.3 Scholars disagree about the Koranic verses and hadiths that deal with the attributes of Allah (n: such as His 'hand' (Koran 48:10), His 'eyes' (52:48), or His 'nearness' (50:16 as to whether they should be discussed in terms of a particular figurative interpretation (ta'wil, def: w6) or not. Some say that they should be figuratively interpreted as befits them (n: interpreting His 'hand,' for example, as an allusion to His omnipotence). And this is the more well known of the two positions of the scholastic theologians. Others say that such verses should not be given a definitive interpretation, but rather their meaning should not be discussed, and the knowledge of them should be consigned to Allah Most High, while at the same time believing in the transcendence of Allah Most High, and that the characteristics of created things do not apply to Him. For example, it should be said we believe that "the All-merciful is 'established' [Ar. istawa, dis: v1.3] on the Throne" (Koran 20:5), but we do not know the reality of the meaning of that, nor what is intended thereby, though we believe of Allah Most High that "there is nothing whatsoever like unto Him" (Koran 42:11)
These ayat are:
Indeed, those who pledge allegiance to you, [O Muhammad] - they are actually pledging allegiance to Allah. The hand of Allah is over their hands. So he who breaks his word only breaks it to the detriment of himself. And he who fulfills that which he has promised Allah - He will give him a great reward. -- Qur'an 48:10
And We have already created man and know what his soul whispers to him, and We are closer to him than [his] jugular vein -- Qur'an 50:16
And be patient, [O Muhammad], for the decision of your Lord, for indeed, you are in Our eyes. And exalt [ Allah ] with praise of your Lord when you arise. -- Qur'an 52:48
The Most Merciful [who is] above the Throne established. -- Qur'an 20:5
These are problematic to take literally as, as Mufti Muhammad ibn Adam at Darul Iftaa writes:
...if one clearly believes that Allah is ‘physically’ in a location or that He has organs – such as hands, feet, face, etc – that are similar to His creation, or one gives Allah attributes of created things, then that would entail disbelief.
So there are things in the Qur'an we cannot take 100% literally. The Darul Iftaa fatwa describes ways of reconciling this:
For example, Allah’s attribute of ‘yad’ has been mentioned in various texts of the Qur’an and Sunna. ‘Yad’ linguistically, as we understand it, refers to the hand of a created being. However, all the groups agree and emphatically deny that Allah has a hand like that of a human, thus they all preserve the central belief in Allah’s transcendence. Thereafter, whether we say “Allah knows best what ‘yad’ means” or “it refers to Allah’s assistance, etc” or “it means a hand but certainly unlike the human hand”, it does not undo the central aqida outlined in the verse, “There is nothing whatsoever like unto Him.” (Qur’an 42:11)
If we take an ultra-literal approach, we're going to run into problems. This implies that it's reasonable to have non-literal interpretations the Qur'an in some cases. However, the Darul Iftaa fatwa also points out:
the following positions... may well even take one out of the fold of Islam: ... Rejecting and denying the non-decisive (mutashabihat) texts concerning the attributes of Allah altogether. This is known as ta’til.
So there are things which are in the Qur'an are intended to be taken literally, while other things should not be taken too literally (and we were warned about this in Qur'an 3:7).