To address the original question,Why are the Eagle, the Ox, the Man and the Lion used continuously as symbols for the different clans of Jews, the different books of the New Testament, etc.
Ezekiel 1:10 reads of the Cherubim in his vision: "Their faces looked like this: Each of the four had the face of a man, and on the right side each had the face of a lion, and on the left the face of an ox; each also had the face of an eagle."
The cherubim are the original source for this grouping of 4. It is possible that the cherubim on the Mercy Seat of the Ark were thus designed (we cannot know for sure, since Ezekiel's vision was way after the time of Moses; did Moses perhaps know how cherubim looked? Do all cherubim look this way?).