Hello Vic,
There were many stories associated with that figure, in one of them, it was the lord of the State of Qi (齐) escaping his enemies after being defeated. Just when he came to a river and had nowhere to run, a big bird flew and carried him to safety. The reasoning for putting the figure at the very edge of the spine of the roof structure was to imply being saved from danger (逢凶化吉).
Another version tells that a tyrant prince in 3rd century b.c. was defeated and killed by his enemies. To remember his evil deeds the people put his figure on a hen on the end of the roof. The hen cannot jump from the roof, because the prince is too heavy and the hen cannot go back, because a dragon is on the other end.. Between the princes figure and the dragon are some other figures. All figures are symbols to prevent the house from fire and lightning
That is exactlly the picture you showed .Those animals were also said to be aesthetic coverings for the reinforcing nails that were necessary where sections of the roofs meet, though each animal had a mythical connotation.
The dragon represents the state.The man riding the chicken represents a minion of the emperor .The Mythical beast behind the man is set to devour him should he stray from performing his duties to the empire with faithfulness and rectitude. The number of such beasts ( always an odd number) indicates the importance of the duties performed within that building.
Other superstitious stories say that these statues were there to keep the bad ghosts away.