Are the aforementioned Hadith and supplication authentic, and is it permissible to say them in prostration or during the Tashahhud? And would saying them be considered an innovation if they are not authentic?

The aforementioned hadith is weak, but the supplication contained within it has been authenticated in meaning in another hadith from Aisha (may Allah be pleased with her) with a longer wording, where she asked the Prophet (peace and blessings be upon him): "O Allah, I ask You for all good, both immediate and delayed..." It is permissible to supplicate with any good, appropriate, and soundly meaningful supplication, even if it is in a weak hadith or has not been reported in an authentic narration. A servant may choose in their prayer whatever supplication pleases them for the good of this world and the Hereafter, but observing the supplications authentically reported from the Prophet (peace and blessings be upon him) is more preferable.