Is the Friday Prayer complete for one who catches the last Tashahhud with the Imam, by completing it as four rak'ahs like the Dhuhr prayer, or is it valid for a congregant who catches one rak'ah with the Imam to complete it with one rak'ah, which would be counted as Friday Prayer?

If you catch the Imam in the final tashahhud of the Friday prayer, you have missed the Friday prayer. You should have sought another mosque to catch it. If you were unable to do so, then you have missed the Friday prayer according to the majority of scholars. As for following someone who joined late (masbuq) and caught one rak'ah with the Imam, there is a difference of opinion on this, and it is preferable to avoid it to steer clear of disagreement. It is better to wait for him until he prays alone, then you can pray with him to attain the virtue of congregational prayer. If you do not hope to find another congregation, then joining the Imam to catch the remainder of the tashahhud is better than leaving.