Is it necessary for the worshipper to keep the index finger of the right hand raised after completing the Tashahhud and the prayers upon the Prophet (peace be upon him) in the final Tashahhud, or is it permissible to lower it before the Imam says the Taslim?

It is from the Sunnah to supplicate after finishing the prayer upon the Prophet (peace be upon him) in the last tashahhud, especially seeking refuge from the torment of Hell, the torment of the grave, the trials of life and death, and the evil of the Antichrist (Al-Masih Ad-Dajjal). This is along with continuing to supplicate with whatever you desire for the good of this world and the Hereafter, while raising the index finger until you finish the prayer with your Imam.

As for raising the index finger during the tashahhud, it is among the agreed-upon Sunnah acts of prayer. It is preferable to raise and move it throughout the entire tashahhud while supplicating, as it repels Satan and negates forgetfulness.