Is the prayer of one who intended to pray Jumu'ah and joined it after the Imam rose from bowing, then completed it as four rak'ahs, thinking it had become Dhuhr, valid? And must he repeat it? And is he sinful for his ignorance?

One catches the Friday prayer by catching a complete ruk'ah. Whoever joins the prayer with the intention of performing Jumu'ah and catches less than a complete ruk'ah, he completes it as Dhuhr after the Imam's taslim (definition), while maintaining his original intention. This is considered valid and people have no choice but to act upon it, as Dhuhr is a branch of Jumu'ah, and both are the obligatory prayer of the time.