What is the ruling on someone who catches up with the Imam in the last Tashahhud? Is he allowed to supplicate after the Tashahhud until the Imam says the Taslim, and then complete his prayer? And does his prayer become invalid if he does that out of ignorance?

Whoever catches the last Tashahhud (definition) with the Imam should recite the Tashahhud and supplicate thereafter until the Imam says the taslim (definition). This does not invalidate his prayer. Some scholars are of the opinion that he attains the reward of congregational prayer.