Q:

Confusion about Qur'an Qur’an 33:37 and 33:4-5 and Adoption ruling? 

We know that in Islam adoption is not allowed Qur'an verses Qur’an 33:37 and 33:4-5 pertain to that.  My Question is which of these verses came first, 33:4-5 or 33:37 in terms of order of revelation!?!  Did adoption become haram after Prophet Muhammad s.a.w married Zaynab or before in 33:4-5

A:

Adoption is Haram or not, I am not sure about that but The Verses that you are mentioning, these verses are not revealed for the prohibition of adoption.  They are revealed to abolish some pre-Islamic rituals in Arab..
33:4

Allah has not made for a man two hearts in his interior.  And He has not made your wives whom you declare unlawful your mothers.  And he has not made your adopted sons your [true] sons.  That is [merely] your saying by your mouths, but Allah says the truth, and He guides to the [right] way. 

So Allah says that

he has not made your adopted sons your [true] sons

This was revealed concerning Zayd bin Harithah, may Allah be pleased with him, the freed servant of the Prophet .  The Prophet had adopted him before prophethood, and he was known as Zayd bin Muhammad.  Allah wanted to put an end to this naming and attribution because in the same Surah, Allah is declaring that
33:40

Muhammad is not the father of [any] one of your men, but [he is] the Messenger of Allah and last of the prophets.  And ever is Allah , of all things, Knowing. 

33:5

Call them by [the names of] their fathers; it is more just in the sight of Allah .  But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you.  And there is no blame upon you for that in which you have erred but [only for] what your hearts intended.  And ever is Allah Forgiving and Merciful. 

The prohibition is about consider your adoptive son your real son.  As it is clear in this Ayah, that you cannot call adoptive son with your name.  Call them with their real father name.  and if you don't know their father name, then they are your brother according to this Hadith too..
and
33:37

And [remember, O Muhammad], when you said to the one on whom Allah bestowed favor and you bestowed favor, "Keep your wife and fear Allah ," while you concealed within yourself that which Allah is to disclose.  And you feared the people, while Allah has more right that you fear Him.  So when Zayd had no longer any need for her, We married her to you in order that there not be upon the believers any discomfort concerning the wives of their adopted sons when they no longer have need of them.  And ever is the command of Allah accomplished. 

In pre-Islamic history, if a person adopted a son, they used to regrad him their own offspring, he(adopted son) had right in inheritance. He can not marry the widow of his adopted father nor to the daughter of his adoptive father and if he died or divorce his wife, then his adopted father used to regarded the women as his real daughter-in-law and he could not marry her. 
These rituals were totally against the laws of marriage and divorce and inheritance declared by Allah in Surahs Al-Baqarah and An-Nisa. 
These rituals could not be rooted out by just passing legal order.  So Allah married his Prophet (peace be upon Him) with Hadrat Zainab (RA) the Wife of his adopted son Zaid bin Harithah (RA) after Zaid had divorced her. 

For more information regarding this, please follow:

 Ibn-Kadhir refereance.