Battle of Karbala
I am not aware of any classical sources that specifically identify Za'far the jinn as coming to Karbala, although this story became popularized later. However, there are narrations which indicate that some of the jinn sympathized with or sought to aid Imam Husayn (A), and so the story of Za'far the jinn can be seen as reflective of that general idea.
Yes, this has been mentioned in many books.
His name in Arabic us زعفر Za'far.
Books of History mention that Hazrat Abul Fazl Al-Abbas (a.s.) had 5 sons;
2. Al Fazl,
3. 3. Al Hasan,
4. 4. Al Qasim and
5. 5. Muhammad.
6. History also mentioned that he had 2 daughters. Mohammad Ibn Al-Abbas is mentioned as one of the Martyrs in Karbala.
Obaydullah son of Al-Abbas Ibn Ameerul Momeneen (a.s.) is been mentioned as the son who left children. All the grandchildren of Al-Abbas (a.s.) were from the sons of Obaydullah. The grand and great grand children of Al-Abbas (a.s.) are many and they include many Scholars and very pious Momeneen and very respected Personalities in the history of Muslims. Of the well-known personalities from the great grandchildren of Al-Abbas is Al-Hamza who is buried in Al-Midhatiyyah near Al-Hilla and the whole area is now is called after him as Al-Hamza.
Some of the martyrs were like that before Karbala like Zuhair ibn al Qayn who was previously a supporter of Othman ibn Affan, but after listening and following Imam Hussain (AS), he believed fully in Ameerul Mo'mineen and Imam Hasan and Imam Husain (AS), that is why he sacrificed his life in the way of Allah under the divine leadership of Imam Husain (AS).
Human beings have their own ways to express their feelings of grief and sadness as well to express their feelings of happiness. Islam allows different ways of expression of feelings as far as it does not contradict a clear ruling of Islam. No one can express his happiness by dancing, simply because dancing is forbidden in Islam. No one can express his sadness by drinking beer or alcohol, simply because drinking alcohol or beer is forbidden in Islam.
People whom you mentioned in your question are expressing their feelings of grief by their own way which does not contradict with any Islamic clear rule, and does not have bad effect on the image of Islam.
Allah (SWT) grants people the reward according to their intentions. These people are observing and remembering the tragedy of Karbala. May Allah (SWT) grant them reward according to their intentions.
B) Obaydullah son of Al-Abbas Ibn Ameerul Momeneen (a.s.) is been mentioned as the son who left children. All the grandchildren of Al-Abbas (a.s.) were from the sons of Obaydullah.
Books of History mention that Hazrat Abul Fazl Al-Abbas (a.s.) had 5 sons. Obaydullah, Al Fazl, Al Hasan, Al Qasim and Muhammad. History also mentioned that he had 2 daughters. Mohammad Ibn Al-Abbas is mentioned as one of the Martyrs in Karbala.
Obaydullah son of Al-Abbas Ibn Ameerul Mo'mineen (a.s.) is been mentioned as the son who left children. All the grandchildren of Al-Abbas (a.s.) were from the sons of Obaydullah. The grand and great grand children of Al-Abbas (a.s.) are many and they include many Scholars and very pious believers and very respected personalities in the history of Muslims. One of the well-known personalities from the great grandchildren of Abul Fazl Al-Abbas (AS) is Al-Hamza al-Sharqi who is buried in Al-Midhatiyyah near Al-Hilla in Iraq and the whole area is called after him as Al-Hamza al-Sharqi.
According to Ibn Tawoos in al-Iqbal, the wife of Hazrat Abbas, Lady Lubabah Bint Obaidullah Ibn Abbas, was in Karbala and she witnessed the tragedies and was taken as captive with other ladies and children of Ahlul Bayt (AS) to Kufa then Shaam. She was crying day and night in the tragedies of Karbala and passed away in Madina shortly after returning back. Her children were then looked after by their grandmother Lady Ummul Banin.
There are some narrations indicating that the believer (mu'min) and/or those who have allegiance (wilayah) to Ahl al-Bayt (A) are created from a specific sort of clay, which is sometimes associated with the Euphrates. The faithless (kafir) and/or enemies of Ahl al-Bayt (A) are created from another sort of clay. And most people are of a sort of mix between the two which is why we have diverse temperaments.
Anyway, if these narrations are correctly transmitted, they seem to be more metaphorical about our natures and not literally relating to what kind of components we are physically constructed from. In fact, given that we human beings primarily gain our physical material from eating, and food is imported and shipped worldwide today, we are often built from the "clay" of many different regions!
So with that in mind, it is reasonable to say that someone with a strong affinity for Ahl al-Bayt (A) might have some creational link to Karbala metaphorically or metaphysically.
That being said, I am not aware of any text that indicates that we are buried in the same place whose clay we are created from, or which really assigns any significance to the region where we are physically buried.
In fact that Qur'an tells us that we don't know what land we will be buried in; life is full of surprises and we never know where we will go, nor when we will go.
Similarly the Qur'an does not assign any particular ethical significance to where we live and just says that Allah's earth is vast and if we are unable to live freely in one area, we should move.
Regarding whether he had a daughter named Fatima in Madina, there is some discussion here: https://www.al-islam.org/ask/what-information-is-available-on-the-life-o...
Historians do not agree on how many children Imam Husain had. However, this is a good summary of what various authors have said: https://en.wikishia.net/view/Imam_al-Husayn_b._%27Ali_(a)#Wives_and_Children
Hope that helps - history is a challenging subject!
Allah (SWT) does not enforce justice in this life stopping people from doing wrong, but He gives freedom and leaves human beings to select their own options, whether good or bad, although He guides and helps them to do good, that is why He did not stop Cane from killing his brother Able and did not stop the criminals who killed prophets like Yahya (AS) and did not stop Ibn Muljam from killing Imam Ali (AS). That is the Justice of Allah which gives every one the result of his own actions.
Traditionally, it is said that Sukayna bint al-Husayn and Ruqayyah bint al-Husayn were 2 daughters of Imam Husayn who were present in Karbala. (In some cultures, they are treated as one and called "Sakina").
From a historical perspective, it can be deduced from historical reports that Sukayna bint al-Husayn was a young woman (maybe about 13 years old) at the time of Karbala, and that this is the same Sukayna bint al-Husayn who was mentioned in later historical sources. Also, in a poem ascribed to Imam Husayn to comfort her, he refers to her as the "best of women" (that is, she wouldn't have been a young child or else he wouldn't have called her a "woman"):
سيطول بعدي يا سكينة! فاعلمي * منك البكاء إذا الحمام دهاني لا تحرقي قلبي بدمعك حسرة * ما دام مني الروح في جثماني وإذا قتلت فأنت أولى بالذي * تأتينه يا خيرة النسوان
"O, my dear Sukayna! Know that after me your weeping is prolonged. [So my daughter] do not burn my heart by your sorrowful tears as long as I am alive. O, the best of women! Weeping is more suitable for you after my martyrdom."
It is said that Ruqayyah was about 3 years old in Karbala, that she died when Yazid sent her the head of her father, and that her shrine is in Syria. However, historical reports about her are scant. Maybe people in those days didn't consider it too important to keep records about young children as children often did not live past childhood.