How many rak'ahs are there in Salat al-Tahajjud, is it permissible to supplicate for the good of this world and the Hereafter in prostration and after concluding the obligatory prayer with salām, and is Salat al-Istikharah performed in the last third of the night considered Tahajjud?

Tahajjud is a supererogatory prayer performed after sleep. The minimum number of its rak'ahs (units of prayer) is two light rak'ahs, by scholarly consensus. As for the maximum number, there is disagreement:

Hanafi school: eight rak'ahs. Maliki school: ten or twelve rak'ahs. Shafi'i school: there is no limit to the number of its rak'ahs.

Supplication during prostration is prescribed and is one of the occasions when prayers are answered. As for supplicating for worldly pleasures during prayer, it invalidates the prayer according to the Hanafi and Hanbali schools, but it is permissible according to the Shafi'i and Maliki schools.

Supplication after the obligatory prayers is prescribed, and one may supplicate for the good of both this world and the Hereafter therein.

The time for Tahajjud begins in the last third of the night. If you pray Istikharah (prayer for guidance) at that time, it is considered part of Tahajjud, and two rak'ahs from Tahajjud can take its place, followed by the Istikharah supplication.