Participating in the mourning of Imam Hussein (peace be upon him).

The greatest and the most tragic calamity to the friends of Allah and His righteous slaves happened on the day of Ashura. The son of the Messenger of Allah's daughter, his companions and family members including the infant child were killed and their bodies were scattered afterwards. The enemies attacked the tents in which the women and children gathered amongst which were the family members of the Messenger of Allah (peace be upon him).
The Imams of the Ahlulbait (peace be upon them) remembered this day by recounting the tragedy and expressing their sadness and grief. They also ordered their followers to do the same.
They also emphasized visiting the grave of the oppressed Imam Hussein (peace be upon him) on this day or recite the Ziyara if visiting his grave in Kerbala is not an option.

It is permissible to offer the believers such greetings on the tenth of Muharram. It is recommended for the believers to also offer their condolences to each other in the grief of the tragedy of Imam Hussein (peace be upon him) by saying: “Addhamallahu ujoorana bi-Musabil-Husain, wa Ja’alana wa Iyyakum minat-Talibeen bi Tharihim ma’a Waliyyihi al-Imam al-Mahdi min Aali Mohammad.”
The translation: “May Allah greaten your reward because the tragedy of Imam Hussein and make you and us amongst the seekers of his revenge with Imam Mahdi (may Allah hasten his reappearance)”.

Removing the turban is not obligatory but it could be – in some countries and based on certain conventional understandings – an expression of sadness and grief on what had happened to Imam Hussein and the Ahlulbait (peace be upon them) in Karbala when they faced numerous kinds of tragedies like their decapitation and placing their heads on spears.

All what believers do in order to express their grief and sadness and show their loyalty to the Ahlulbait (peace be upon them) are amongst the best of deeds. One should do so in such a way that the person is not harmed with a prohibited kind of harm.

It is not prohibited to study in the day or the night of Ashura.

Engaging in such activities on the night or day of Ashura is not in accordance to what has been narrated from the Imams (peace be upon them) in the exhortation to express grief and sadness over the tragedy suffered by Imam Hussain (peace be upon him), his household and companions on the day of Ashura.

These days are days of sadness and mourning for the Shia because of the tragedies that befell the Ahlulbait in this period of the year.

Yes, it is permissible.