Ziyarat Ashura (Arabic: زیارة عاشوراء) is a Shia salutatory prayer to Husayn ibn Ali and the martyrs of the Battle of Karbala. The prayer is part of the liturgy used in pilgrimages to the shrine of Husayn in Karbala.
Not compulsory. Although it is always recommended to be with Wudhu as much as you can, but it is not obligatory to be with Wudhu for reciting Ziyarat or for reciting Quran or reciting Du'a. It is just recommended and not obligatory.
To add to the answer (even though the answer is already complete in and of itself), the narration through which we receive Ziyarat 'Ashura says: "When the Imam finished reciting the ziyara, he told me 'O 'Alqama, if you can recite this ziyara every day during your lifetime, do it and all its reward will be yours."
[The full part of the narration is as follows]
Alqama b. Muhammad al-Hadrami says: I requested Imam al-Baqir (a) to "teach me something so that I can recite it when I visit him (Imam al-Husayn (a)) from nearby on this day (day of Ashura): and teach me something so that I can recite it when I want to direct my salutation toward him from regions far from Karbala and from my home. The Imam said: O 'Alqama, after you direct salutation toward him, perform a two-rak'a prayer, and after that, recite this ziyara-text, by reciting this ziyara you have truly called him in the way that angels call him when they want to address him, and Allah gives you thousand thousand (one million) rewards and forgives thousand thousand sins of yours and elevates your status thousand thousand levels, and you will be considered as one of those who were martyred along him and you even share their status; and the rewards of the Prophets, Apostles and all those who have visited al-Husayn b. Ali (a) since he has been martyred, would as well be written for you." 'Alqama continues: "When the Imam finished reciting the ziyara, he told me 'O 'Alqama, if you can recite this ziyara every day during your lifetime, do it and all its reward will be yours."
So it is recommended and some people do it but of course it is not required.
Yes. It is highly recommended to recite Ziyarat Aashura every day even on Eid day or birthdays of Imams (AS).
Ziyarat Ashura is one of the most authentic Ziyarat's narrated by most authentic narrators. It was narrated from Imam Mohammad Al-Baqir (AS).
Ibn Qawlawayh narrated Ziyarat Ashura in his authentic book Kaamil al-Ziyarah, page 325 . All his narrators are authentic narrating from Imam Mohammad Al-Baqir (AS). All our prominent scholars mentioned this Ziyarat in their books.
It is highly recommended to perform these two Rak'ats of Salah of Ziyarat but not obligatory.
The narration in Kaamil Al-Ziyarah is more authentic. Nevertheless it is good to recite السلام عليك يا خيرة الله وابن خيرته with intention of Qurbatan Elallah Ta'alaa.
It is permissible to prostrate on a clay on which something is written, though it is much better to prostrate on plain clay ( Torbah) or Mohr or Sajdagah.