Does the prayer become void if "التحيات لله، والصلاة" (All greetings are for Allah, and the prayer) is said in the final Tashahhud instead of "والصلوات" (and all prayers), with the performance of Sujud Al-Sahw (prostration of forgetfulness) after the prayer?

Your prayer is valid and not nullified by using the singular form of "prayer" (الصلاة) instead of its plural in the Tashahhud. While it is more complete to perform the Tashahhud as narrated in the Hadith, it is not a condition for the validity of the prayer as long as the worshipper has performed the sufficient minimum.

Al-Shafi'i stated in "Al-Umm": "And if a man does not add more to the Tashahhud than saying: 'All greetings are for Allah, I bear witness that there is no god but Allah, and I bear witness that Muhammad is the Messenger of Allah. Peace be upon you, O Prophet, and the mercy of Allah and His blessings. Peace be upon us and upon the righteous servants of Allah.' And he invoked blessings upon the Messenger of Allah, I would dislike that for him, but I would not consider it necessary for him to repeat [the prayer], because he has brought the name of Tashahhud, and blessings upon the Prophet (peace be upon him), and peace upon the Messenger of Allah (peace be upon him), and upon the servants of Allah."

Ibn Qudamah said in "Al-Mughni": "And by whichever Tashahhud one performs out of those authentically narrated from the Prophet (peace be upon him), it is permissible. Ahmad explicitly stated this, saying: 'Abdullah's Tashahhud is more pleasing to me, but if one performs it with another, it is permissible; because when the Prophet (peace be upon him) taught the Companions - meaning he taught them the Tashahhud - with variations, it indicates the permissibility of all of them... Al-Qadi said: This indicates that if one omits a word that is omitted in some narrated Tashahhud forms, their Tashahhud is valid. Based on this, it can be said: The minimum sufficient is: 'All greetings are for Allah, peace be upon you, O Prophet, and the mercy of Allah, peace be upon us and upon the righteous servants of Allah, I bear witness that there is no god but Allah, and I bear witness that Muhammad is His servant and Messenger,' or 'that Muhammad is the Messenger of Allah.' Ahmad, in a narration by Abu Dawud, said: If he says, 'and that Muhammad is His servant and Messenger,' and does not mention 'and I bear witness' - I hope it will suffice him."