To what extent are the hadiths valid that mention the virtue of performing a two-rak'ah prayer on Friday night, reciting Surat Az-Zalzalah 15 times, and a sixteen-rak'ah prayer on Saturday night, reciting Surat Al-Ikhlas 31 times?

There is no authentic hadith (definition) regarding the virtue of singling out a specific day or night of the week for a supererogatory prayer or vigil. Everything reported in that regard is repudiated and fabricated against the Messenger of Allah, peace and blessings be upon him, and it is not permissible to narrate or act upon it. Whoever does so has innovated in religion. The people knowledgeable in hadith have unanimously agreed that the hadiths reported concerning the prayers of Sunday, Monday, and all other days of the week are fabricated, and none of the imams of the religion deemed them desirable. The Prophet, peace and blessings be upon him, forbade singling out Friday night for vigil or Friday day for fasting. Ibn al-Qayyim, al-Iraqi, and al-Shawkani mentioned that the hadiths about the prayers of specific days and nights are false, weak, and repudiated.