Is a latecomer who erred in the intention of sitting between the two prostrations and sitting for the Tashahhud required to perform the prostration of forgetfulness, and is he obliged to repeat the prayer if a long time has passed since it?

The congregant's recitation of a supplication before the Tashahhud does not affect the validity of the prayer. Rather, it is a prescribed remembrance performed out of its proper place. This does not necessitate a prostration of forgetfulness (sujud al-sahw), because the Imam bears the forgetfulness of the congregant. Even if we assume it was an act of forgetfulness, it does not mandate a prostration of forgetfulness.