I'Tikāf

wa alaykum as salaam wa rahma!

Thanks for your questions! Here is a portion of a quotation from the Ahkaam of Imam al-Haadi (as) regarding religious seclusion (al-i'tikaaf):

Religious seclusion is when a man intends and stipulates a period of seclusion for particular days or says: “It is for Allah that I seclude myself such and such days.” If he obligates that upon himself by stipulating his intention or by the aforementioned wording, he is to enter the mosque at the beginning of the time he stipulated upon himself and fast those days that he intended. Consequently, there is no religious seclusion without fasting.
Then, he is to be on guard from obscenity, lying, argument, debate and other forms of corrupt acts and speech. The one in religious seclusion is to increase in recitation of Qur’an, remembrance, seeking forgiveness and glorification of the Most Gracious.
He does not leave the mosque except for the reasons we mentioned: to fulfil a need, tend to one of the Muslims, follow the funeral bier of the believers, and in the case of a need to command or prohibit his family. He is to command and prohibit them while standing and not sit down before returning to the mosque.

Religious seclusion during Ramadan is encouraged due to it being consistently practiced by Allah's Messenger, peace and blessings be upon him and his progeny. 

As for ahadith regarding its virtue, most of our narrations only mention our Prophet, peace and blessings be upon him and his progeny, performing it. Our following him in this act of worship should be sufficient as a virtue. 

Hope this helps! And Allah knows best!

IRS 

Thank you for your question! Yes, religious seclusion is not restricted to one gender; women can observe it as well. Imam al-Haadi (as) says the following in the same book:

If a woman obligates herself to observe religious seclusion for two months— more or less—she is obligated to observe what she obligated upon herself. While she is in religious seclusion, she is obligated to fast just as the man. That which invalidates it for her is the same as that which invalidates it for him, and that which makes it sound for her is the same as that which makes it sound for him. If she menstruates while in seclusion, she is to depart from the mosque. And once she is pure and has purified herself, she is to return to her religious seclusion. She then observes religious seclusion and the fast she obligated upon herself from the time she broke her fast until she completes the days she appointed for Allah.

Any person in religious seclusion—whether male or female—who fears for himself or herself in the mosque of their seclusion must depart that mosque and enter another. 

And Allah knows best!

IRS

As for a hadith related to the virtue of religious seclusion, I found this narration in the Amaali Ahmed b. 'Isa (as) in which he reports a hadith on the authority of Sayyid Imam Ja'far b. Muhammad as-Saadiq from his forefathers, upon them be peace, that the Prophet, peace and blessings be upon him and his progeny, said: ((Whoever performs religious seclusion on the last ten nights of Ramadan will be considered the same as one who performed two Hajj pilgrimages and two 'Umra pilgrimages)).