I don't know the context of this event, I don't know whether it really happened, and I don't know whether it happened before or after marriage, but some aspects of the letter seem questionable to me. As for the content of the rules, some of the rules are valid in Islam, while some of them seem to have glaring issues, based on the scholarly views I know of.
1- “You will wear hijab at all times.”
There are multiple classes of people in front of which a woman is generally not obligated to wear hijab in Islam. “And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed.”(Qur’an 24:31) It seems to me that demanding her to constantly wear it would probably make great difficulty and restriction for her. I doubt the husband has any need for her to wear hijab at all times, and if he is telling her to do so just so he can feel good about himself, or another non-Islam-based reason, it seems to me to be blatant unkindness or at least grounds for the wife to get a separation, if he made her life unbearable with it.
2- "You will not work, even if I am not able to financially support you..." According to scholarly views, if the man cannot financially support the woman, she is not obligated to obey him, she does not have to be intimate with him, she can go out and work without his permission, and she can get a separation. “Ash-Shirazi ash-Shaafi‘i said in al-Muhadhdhab fi Fiqh al-Imam ash-Shaafi‘i (3/155): If she chooses to stay after he becomes unable to spend on her, she is not obliged to allow him to be intimate with her, and she may leave his house, because allowing intimacy is in return for maintenance, so it is not obligatory when there is no maintenance. End quote. Al-Bahooti (may Allah have mercy on him) said in Kashshaf al-Qina‘ (5/477): She has the right to remain in the marriage and not allow him to be intimate with her, so she does not have to allow intimacy or reside in his house, and he must not detain her; rather he should let her earn money, even if she is well off, because he has not given her anything in return for intimacy with her. End quote.”(source: translation from islamqa.info)
3- “unless you serve him food or drink.”
That makes it sound like he wants his wife to serve a stranger man, which sounds like potential unnecessary mixing. Why can’t he do it himself?
4- "I am allowed to beat you softly, but only under extreme circumstances, such as denying sex."
The way this is written makes it sound like the man would beat her just for denying sex. In Islam, there are steps to take before the "beating"(which according to scholars, must not cause pain). These steps are first advising the wife and then forsaking her in bed. I don't see those mentioned in the husband's letter. He could have said "extreme circumstances, such as when advising and forsaking don't do anything", but instead he said "extreme circumstances, such as denying sex", which makes it sound like he considers denying sex on its own to be a circumstance under which beating would be permissible. That is just how it sounds to me. Again, I do not know the context of this event. However, if the wife did nothing wrong, and the husband simply handed her that list, it would feel to me that he is violating her right to be treated kindly by speaking to her in an unkind tone. At least, if I were the wife, I would feel treated unkindly. However, none of this is a reason to find fault with Islam. Islam honors and privileges women and protects their rights.
These are issues with the rules that stood out to me; I'm not knowledgeable enough to say for certain if there are other issues with them.
Another aspect of the letter, however, stands out. One of the rights of the wife is to be treated kindly, according to scholars.
“O you who have believed, it is not lawful for you to inherit women by compulsion. And do not make difficulties for them in order to take [back] part of what you gave them unless they commit a clear immorality. And live with them in kindness. For if you dislike them - perhaps you dislike a thing and Allah makes therein much good.”(Qur'an 4:19, interpretation of the meaning)
“And live with them in kindness.” [4:19]
Ibn Katheer said: i.e., speak kindly to them, treat them well and pay attention to your deeds and your appearance as much as you can; as you would like her to do for you, do the same for her.”(source of translation: islamqa.info)
Allah said in another ayah, وَلَهُنَّ مِثْلُ الَّذِى عَلَيْهِنَّ بِالْمَعْرُوفِ (And they have rights similar over them to what is reasonable) 2:228.
It was narrated that Abu Hurayrah (may Allaah be pleased with him) said: "The Messenger of Allaah (peace and blessings of Allaah be upon him) said: 'Be kind to women.'"(Narrated by al-Bukhaari, 3153; Muslim, 1468).(Source of translation: islamqa.info)
Abu Shuraih Khuwailid bin 'Amr Al-Khuza'i (May Allah be pleased with him) reported: The Prophet (sallallahu alayhi wa sallam) said, "O Allah, I declare inviolable the rights of two weak ones: the orphans and women".(source of translation: sunnah.com)
“The Messenger of Allaah (peace and blessings of Allaah be upon him) say: “The worst of guardians are those who are cruel. Beware lest you be one of them.” Narrated by Muslim (1830)”(Source of translation: islamqa.info)
Haritha ibn Wahb reported: The Messenger of Allah, peace and blessings be upon him, said: أَلَا أُخْبِرُكُمْ بِأَهْلِ الْجَنَّةِ كُلُّ ضَعِيفٍ مُتَضَعِّفٍ لَوْ أَقْسَمَ عَلَى اللَّهِ لَأَبَرَّهُ أَلَا أُخْبِرُكُمْ بِأَهْلِ النَّارِ كُلُّ عُتُلٍّ جَوَّاظٍ مُسْتَكْبِرٍ Shall I not tell you about the companions of Paradise? They are every humble person considered weak, but if they gave an oath by Allah it would be fulfilled. Shall I not tell you about the companions of Hellfire? They are every harsh, haughty, and arrogant person. Source: Sahih Bukhari 4634
Ibn Abbas (may Allah be pleased with him and his father too) said: “I like to beautify myself for my wife as I like her beautify herself for me; as Allah said: {and live with them honourably}.”(Source of translation: islamqa.info)
The tone of the letter of this supposed husband, in multiple places does not sound kind, in my opinion.
For example:
1- "Here are my demands..." instead of more gentle word choice.
2- Prefacing every rule with "You will", rather than something more considerate in tone.
3- "Women are weak, you are weak..."
Apart from the insulting-sounding usage of this phrase here, the truth is that every human is weak in some way. I am not aware of any Islamic daleel to back a blanket statement that women are weak, as opposed to men, with regard to sexual temptation. I do know of this ayah and some of its tafaseer.
"Allah wishes to lighten (the burden) for you; and man was created weak” [an-Nisa’ 4:28].
“Ibn al-Jawzi (may Allah have mercy on him) mentioned three views concerning the verse, as he said:
With regard to what is meant by human weakness, there are three views:
1-That it refers (in general terms) to man’s inherent weakness. Al-Hasan said: It (refers to the fact that) he was created from semen of worthless water (32:8).
2-That he is weak and lacks patience in the face of the temptation of women. This was the view of Tawoos and Muqaatil.
3-That he is weak in resolve when overwhelmed by emotions and desires. This is the view of az-Zajjaaj and Ibn Keesaan.
End quote from Zaad al-Maseer (1/395)”(Source of translation: islamqa.info)
4- "so if you were a good Muslim you will even help me search for a wife"
This sounds as if the husband is criticizing the wife's piety, or using it to guilt her into feeling like it's her duty to help him look for a wife, or trying to make her think her feelings about it don't matter to him in the least. I'm not aware of a daleel that a wife should help her husband look for another wife, nor a daleel that bases a woman's piety on whether she does that for her husband. I have, however, seen the scholarly view that the wife is entitled to her own house with the husband, separate from any other wives or the husband's family.
“Al-Kaasaani (may Allaah have mercy on him) said: If a husband wants to make her live with a co-wife or her in-laws, such as his mother or sister or daughter from another marriage or anotherrelative, and she refuses to accept that, then he has to provide her with accommodation of her own… But if he lodges her in a room of the house that has a door of its own, this is sufficient for her and she should not ask him for alternative accommodation, because the harm caused by fear for her belongings and not being able to relax is no longer there. (Badaa’i’ al-Sanaa’i’, 4/23)
Ibn Qudaamah said: A man does not have the right to make two wives live in the same dwelling without their consent, regardless of whether the house is large or small, because this will cause them harm due to the enmity and jealousy between them. Making them live together will cause conflict and each of them will be able to hear when the husband spends time with (has marital relations with) the other or she will see that. If they both agree (to live together in one house), this is permissible because they have the right to do to ask for independent accommodation, or they may choose to forgo this right. (al-Mughni, 8/137) (Source of translation: islamqa.info)
Al-Haskafi (may Allaah have mercy on him) – one of the Hanafis – said: Similarly, she is entitled to a place in the house that is free of his family and her family according to their means, as is the case with food and clothing. A separated part of the house with a door of its own and facilities such as a bathroom and kitchen will be sufficient for the intended purpose. Ibn ‘Aabideen commented: What is meant by “a bathroom and kitchen” is bathroom facilities and a place for cooking that should be within the room or in a place which is not shared by any other family members. (al-Durr al-Mukhtaar, 3/599-600)”(Source of translation: islamqa.info)
And the view that women may put in the marriage contract that the man may not marry a wife other than her, as a condition of marriage, and that the man must follow that.(Source of translation: islamqa.info)
“Ibn Qudaamah (may Allah have mercy on him) said: If he stipulates that he will not take her out of her house or her city, or that he will not travel with her or will not take another wife, then he is obliged to fulfil that, and if he does not do so, then she has the right to annul the marriage. This was narrated from ‘Umar, Sa’d ibn Abi Waqqaas and ‘Amr ibn al-‘Aas (may Allah be pleased with them). End quote. Al-Mughni, 9/483”(Source of translation: islamqa.info)
I’ve read ahadith and texts showing how the Prophet(sallallahu alayhi wa sallam) and early Muslim men cared for women’s feelings.
Miswar b. Makhramali reported that he heard Allah's Messenger (ﷺ) say, as he sat on the pulpit: The sons of Hisham b. Mughira have asked my permission to marry their daughter with 'Ali b. Abi Talib (that refers to the daughter of Abu Jahl for whom 'All had sent a proposal for marriage). But I would not allow them, I would not allow them, I would not allow them (and the only alternative possible is) that 'Ali should divorce my daughter (and then marry their daughter), for my daughter is part of me. He who disturbs her in fact disturbs me and he who offends her offends me. (Sahih Muslim 2449, source of translation: sunnah.com)
"It was narrated by al-Bukhaari (3110): “I fear lest she be put to trial with regard to her religious commitment.”
According to Muslim (2449): “Faatimah is part of me, and I fear lest she be put to trial with regard to her religious commitment.
Al-Haafiz Ibn Hajar said: This incident occurred after the conquest of Makkah, at which time none of the daughters of the Prophet (blessings and peace of Allah be upon him) was still alive except her; after losing her mother, she had lost her sisters, and giving her cause to become jealous would have exacerbated her grief.
End quote from Fath al-Baari, 7/86 ”(Source of translations: islamqa.info)
I doubt a pious, knowledgeable man would talk to a Muslim wife the way shown in the question, out of the blue.
What seems potentially hurtful in it is the hostile tone, which(with no context) implies ill-will and uncaringness towards the wife, rather than the parts about Islamic rules.
Disclaimer: As with all my answers, I am not giving a fatwa. I share what I know of Islamic scripture, historical events and words of scholars, and translations of them, and sometimes my opinion, and they may or may not be the correct answer to the question, and I may or may not have interpreted them correctly as being relevant. Scriptural texts posted in English are a translation from the original Arabic and may not encompass the full meaning of the original.