I came home to Lakeview manor to find an effing giant in my animal compound. It had slaughtered the cow and two of the three chickens.

My steward gave me the option to buy another cow, but not replace any of the dead chickens, so I have two chicken corpses inhabiting the compound now.

ditto
you can only replace the cow or horse
the carriage guy is immortal

Same thing happened to me but with Bandits, they killed all my Chickens and Horse. I can replace the Horse, but there is no option to replace the Chickens

Sad, but what do you need the chikens for?

For the sole purpose of having chickens that and i think if you have chickens, eggs will respawn in the little nest.

Well no, eggs do not respawn in the nest and as far as roleplaying is concerned if I was the dungeon master I'd rip you off all exp for wanting that badly to having chickens at your manor as the Dohvakin