My question is, do I have to pray at midnight and over to be considered as Qyam-allail? If just I wake up at night and do some Tasbeeh, is this also as Qyam-allail? If so, do I have to be in a full Tahara (purification) Wudu?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Optional night prayer is a confirmed Sunnah and it is the best prayer after the obligatory ones. The Prophet said: "The best prayer after the obligatory prayer is the optional night prayer.'' [An-Nasaa'ee] It is not a condition for the night prayer to be performed after the middle of the night. Indeed its time starts after ‘Ishaa’ prayer and lasts until Fajr. So whoever prays after ‘Ishaa’, has indeed prayed the optional night prayer. However, the best time for them is the last third of the night as the Prophet said: “Allaah descends every night in the last third of the night [in a way that befits His majesty] and says: Is there anyone invoking Me that I may respond to his invocation? Is there anyone asking of Me so that I may grant him his request? Is there anyone asking My Forgiveness, so that I may forgive him?". [Al-Bukhari and Muslim] Some scholars even consider that the optional night prayer starts from Maghrib.

As regards whether or not remembrance of Allaah is considered as night prayer, the answer is that what is apparent from the narration of the Prophet “The best prayer after the obligatory prayer is the optional night prayer.” What is meant by the optional night prayer is prayer and not simply remembrance of Allaah. There is no doubt that remembrance of Allaah is of great excellence, but it is not a condition to have ablution to do it, though it is desirable. The Prophet said: “I disliked mentioning Allaah except in a state of purity.” [Abu Daawood].

Allaah knows best.