Is it valid to pray six cycles (rak'ahs) in pairs with the intention of Tahajjud, and another six with the intention of Qiyam, or is it more correct to have a single intention, and what should that intention be? And should the opening supplication (Du'a al-Istiftah) be repeated for every two cycles?

Tahajjud is the night vigil (Qiyam al-Layl) performed after sleep, and it does not require a specific intention beyond the general intention of performing Qiyam al-Layl. The manner of Qiyam al-Layl is to pray two rak'ahs, then two rak'ahs. If the worshipper fears the dawn (Fajr), he should pray one rak'ah of Witr. The Prophet, peace and blessings be upon him, used to pray eleven rak'ahs: four, then four, then three. It is recommended to recite the opening supplication (Du'a al-Istiftah) at the beginning of each independent two-rak'ah unit. However, if the worshipper restricts the opening supplication to the first rak'ah of Tarawih, some scholars consider this permissible.