Is the prayer of one who, in the final Tashahhud, did not say "wa rahmatullahi wa barakatuh" after "As-salamu alaika ayyuhan-Nabiyyu (Peace be upon you, O Prophet)," and sufficed with saying "As-salamu alaina wa ala ibadillahis-salihin (Peace be upon us and upon the righteous servants of Allah)," permissible? And what is the ruling on his previous prayers?

Your prayers are valid, and you are not required to repeat them. The phrase "Peace be upon you, O Prophet" is mentioned in some versions of the Tashahhud. Not including "and the mercy of Allah and His blessings" does not invalidate the prayer. Some scholars are of the opinion that it is permissible to limit oneself to some phrases of the Tashahhud without all of them. However, it is preferable and obligatory for a Muslim to adhere to one of the complete forms of Tashahhud narrated from the Prophet, may Allah's prayers and peace be upon him.