Karbala (كَرْبَلَاء, romanized: Karbalāʾ; Persian: کربلا) or Kerbala is a city in central Iraq, located about 100 km (62 mi) southwest of Baghdad, and a few dozen miles east of Lake Milh. Karbala is the capital of Karbala Governorate, and has an estimated population of 700,000 people (2015). The city, best known as the location of the Battle of Karbala in 680 CE, or the Mosques of Imam Husayn and Abbas, is considered a holy city for Shi'ite Muslims in the same way as Mecca and Medina.
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.
SHAFAQNA – Imam Sajjad (AS) had severe stomach ache in Karbala to an extent that he could not carry weapons or fight [1,2,3]. The illness started in Karbala and not from the time Imam Hussain (AS) started his journey towards Iraq. Of course his illness was a divine plan so that one child would remain from Imam Hussain (AS) to undertake the Imamate and the leadership of Muslims.
 Basaerul Darajat, Saffar Qomi, Pages 148 and 163.
 Alershad, Sheikh Mofeed, Page 114.
 E’alammul Wara, Tabarsi, Page 246.
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Further explanation: In some cultures, dating back to ancient or mediaeval times (including parts of Europe/Britain as well as Greece), it was considered inappropriate or immodest for respectable women to ride horses in the normal way, especially if they were wearing long dresses that might not stay in place, so women were expected to ride side-saddle or, as Sayyed mentions, in a litter atop the animal. (I am not saying that no women ever rode horses the normal way, just that this was considered more mannerly.) It seems that women riding horses the normal way may have been seen as suggestive, although, nowadays, in much of the world, there is no stigma attached to it.
If they rode bare horses or regularly saddled horses during that time, it could be taken as a sign of them being treated harshly by the enemy.
Muslim women used to sit on Hawdaj which is like a seat put on the animal and not directly on the animal. Sitting directly on the animal or on saddles is Makrouh for women.
The Prophet (s) used to pray on earth and is also reported to have shown reverence towards the soil of Karbala in his lifetime with the foreknowledge of what would happen upon it.
So although it is not compulsory for the Shia to pray on a turba made from the soil of Karbala, it is considered meritorious to do so.
You can read more details with full references here:
We have in our books that many other sons of Imam Ali (AS) were martyred in Karbala beside Imam Husain (AS) and four sons from Ummul Baneen e.g. Ebrahim ibn Ali, Bakr ibn Ali, Amr ibn Ali, Mohammad al-Asghar ibn Al, Abdullah al-Asghar ibn Ali.
Imam Hasan (A) is said to have had about 15 or 16 children, but there are some historical disagreements about the details of how many children he had or who else [apart from al-Qasim's mother] he also was married to.
Allah (SWT) the Glorious, has saved Imam al-Sajjad (AS) in Kerbala to
keep the Imamate which is compulsory not only for Islam and Muslims
but also for the whole universe. Sadaat are not only the grand
children of Imam Husain (AS) but also the grand children of Imam Hasan
(AS) who are also called Hasani Sadaat. In fact all the Grand Children of Bani
Hashim, the grand-father of the Holy Prophet (SAWA) are called
Well, there are also a lot of descendants of Imam al-Hasan (A) especially in places like Morocco. It is recorded to have been said that if Imam al-Husayn (A) was killed, there would be no grandson of the Prophet (S) alive on the earth anymore, in the sense of the son of the daughter of the Prophet (S). This is true, as Imam Hasan (A) had already been martyred by that time.
However the line of the Imamate went through Imam al-Sajjad (A) so the line of the imamate would have been cut off in that way.
Anyway, it is a good question and demonstrates excellent critical thinking skills!
Many people traveled from Madina with Imam Husain (AS) but their numbers are not exactly known to us because we find in different books of history different numbers.
You can find some details on the website of mam Husain Shrine www.imamhussain.org