Battle of Karbala
Imam Hussain (AS) wanted sincere eye witnesses to tell people after his martyrdom what really happened in Karbala keeping in mind the falsehood which Yazeed and his gang were intending to spread among Muslims against Imam Hussain (AS). Mo'awiyah, father of Yazeed, spread rumors among people in Syria that Ali does not pray Namaz, that is why when the news reached Shaam that Imam Ali (AS) was martyred inside the Masjid of Kufa, people of Shaam started asking: What Ali was doing in the Masjid when he does not pray Namaz?
Yazeed and his gang wanted to fabricate stories against Imam Hussain (AS) to mislead people away from Ahlul Bayt (AS). Imam Hussain (AS) took his family members with him knowing what will happen to him and them, to make them witness and convey the message to all Muslims after him.
The great rule of Lady Zainab and Lady Um Kolthoum after Karbala shows how it was necessary to have eye witnesses to inform Muslim generations and refute the false propaganda of Yazeed.
The sufferings of the family of Imam Hussain (AS) was mentioned by him as narrated when he replied those who asked him why to take his family members: Allah has ordained to see me martyred and see my family members captured شاء الله أن يراني قتيلا ويراهنّ سبايا
The above answer is very accurate.
However, I would like to add that the idea of separation between religion and politics is quite new and really only emerged with the notion of separation of church and the modern nation-state.
During the time of the Prophet (S), especially after the formation of the Muslim community in Medina, the Prophet handled matters that today would be considered both "religious" (like acts of worship) and "political" (like laws and the military).
This continued to be the situation during the early caliphates and early Arab-Muslim empires as well as the times of the Shi'i Imams; for instance, Imam 'Ali (A) being formally appointed as the caliph and the treaty of Imam Hasan (A). That is to say, their role as religious leaders also involved political matters. Conversely, political leaders such as those who took on the caliphate also saw themselves as leaders of Islam.
One can say a similar thing for many other pre-modern empires as well, which were not led by Muslims.
So for that reason it is not really correct to divide the Battle of Karbala into "religious" or "political" since it involved both. There were clearly matters that today would be considered "political" such as succession (that is, it was not a battle over theology) while at the same time, as the previous response emphasized, it was not a ploy for power or this sort of thing. From the accounts of the Battle of Karbala, it is clear that matters both religious and political were discussed between both sides prior to the outbreak of fighting.
Rather one can say it was a religious objection to the use and assignment of political power, and a political response to it (military attack).
Battle of Karbala was never political but it was purely religious to save the religion Islam from the corruption of Bani Umayyad. Imam Hussain (AS) did not aim to gain a post or rule or a political position, but wanted to make Muslims aware the real Islam of the Prophet Muhammad (SAWA) and his Ahlul Bayt (AS) must be followed, not the corrupt rulers like Yazeed and his gang, that is why, he repeatedly declared right from Madina to Makka to Karbala that he is going to be martyred in Karbala to save the religion of Islam as there was no other option to save Islam at all but only through his martyrdom.
Imam Hussain (AS) knew definitely that he will be martyred in Karbala and went for his martyrdom because it was the only way to save Islam from deviation and Muslim Ummah from doing astray.
No doubt, Imam Hussain was informed by his grandfather, Prophet Muhammad (SAWA), his mother (SA) , his father (AS) and his brother Imam Hasan (AS) about details of what will happen in Karbala. Allah (SWT) grants His best servants from His knowledge of the Unseen.
Imam Hussain (AS) knew that he will be martyred in Karbala and went all the way to Karbala to save Islam in his martyrdom as no way was possible to save Islam from the deviation but his martyrdom. Imam Husain's aim was not his personal safety but saving Islam even if the cost was his martyrdom.
He kept telling his family members and companions that he will be martyred.
There are many authentic sources about Karbala tragedy which can be found in the authentic narrations from Ahlul Bayt (AS) in our books of Hadeeth like Al-Kaafi, Al-Tahtheeb, Al-Estibsaar, al-Faqeeh, Jaami' al-Ziyaraat,etc.
Yes. We have authentic Hadeeths that weeping for tragedies of Ahlul Bayt (AS). Imam Muhammad Al-Baqir (AS) said: Any believer who weeps on our sufferings from our enemies till his tears flow on his cheek, Allah will grant him a place in Paradise. Kaamil al-Ziyarat by Ibn Qawlawayh , page 100.
According to Amaali al-Shaikh al-Sadouq, Janab e Muslim ibn Aqeel (AS) had two sons martyred in the day of Ashura with Imam Husain (AS) and another two sons who ran away from Karbala after the martyrdom of imam Husain (AS). These two sons where martyred later on after Karbala and were buried near Musayyab in Iraq. أمالي الشيخ الصدوق
Hazrat Abbas (AS) was the chief commander of the army of imam Husain (AS) and he had to remain till the end of the battle.
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We know that 18 of the sons of Imam Ali (AS) were martyred in Karbala. They are Imam Husain (AS), Abul Fadhl Al-Abbas (AS) and another 16 sons.
Muhammad ibn Al-Hanafiyyah was not well and could not join the caravan of imam Husain in Karbala.
Mecca, Medina, and Karbala are all considered holy sites in Shi'i narrations and tradition.
It is not really useful to try to rank them. Mostly people ask this for sectarian reasons, i.e. to try to show whether or not Shi'is have the "correct" belief for a Muslim, or to judge whether individual Shi'is have the "correct" belief.
Each of these sites has its own role in the history of Islam and also has its own spiritual character.
Furthermore, individual Shi'is vary in their personal views or personal perceptions of spiritual matters, such as their experience of a place.
However, Mecca is central because it is the place for the required hajj, the direction of prayer, the homeland of the Prophet where he began his mission, and the sacred haram which is visited by the angels and sanctified in the Qur'an. It is also the birthplace of Imam 'Ali and is said to be the place where Adam and Eve first came to earth.
Medina is also important as the site of the first Muslim community, the site of the final resting place of the Prophet, and the residence of some of the Imams.
According to Shi'i law, it is necessary (wajib) to perform pilgrimage to Mecca, and according to Shi'i narrations, one should also visit the Prophet's final resting place in Medina. It is considered highly recommended but not shari'ah-wajib to visit Karbala. So this gives some ranking.
It is also narrated from Imam al-Sadiq that every evening, seventy thousand angels descend, circle the Ka'bah, then go to visit the graves of Imam 'Ali and Imam Husayn, and then return to the heavens. So this also gives some idea of ranking.
Karbala is also emphasized in the Shi'i tradition, both for its spiritual merits, and also because the martyrdom of Imam Husayn is seen as central to the survival of Islam.
There are a handful of narrations in which, on a creational or supra-worldly level, Karbala is treated as higher than Mecca. However, there are also narrations from the Shi'i Imams that say things like "Allah has preferred Mecca over all other cities" and "one prayer in the Prophet's Mosque [in Medina] is equal to a thousand prayers prayed elsewhere."
So, from this, perhaps we can understand that the intent in these narrations is not to rank sacred sites, but just to indicate that certain places have a special sanctity.
Today, some Shi'is individually may say that, for them, visiting Karbala was a more spiritual experience than performing the hajj. This is likely due to the regrettable materialism that has overtaken Mecca (the clock tower, McDonald's, luxury hotels and meals for those who can afford it, etc), the demolition or alteration of sacred sites (such as the jamarat or Jannat al-Baqi'), mismanagement, the crowds, and the hostility towards Shi'is in today's Saudi, including from the minbar and by the religious police.
In any case, Shi'is worldwide are dedicated to performing the hajj, visiting Medina, and also visting Karbala.