To kill the goose that lays the golden eggs is an allusion to a fairy story.  It is related to the next phrase:

Don’t kill the golden goose.
Obviously it is not a good idea to cut off the source of income.

As happy as a lark is a synonym from nature.  A lark flies high into the sky singing as it goes.

Parrot-fashion means speech without thought.  Parrots and some other species of bird may be taught to speak, but perhaps not to think!

As the crow flies is the shortest distance.

Higher up the pecking order refers to the behaviour of birds.  Whether it is a group of poultry or wild birds feeding, some individuals are always first to the food, which they peck at.  There is a hierarchy in any society.

Running around like a headless chicken is a bit of a gory picture.  If the head is cut off a hen, its muscles still work by reflex and it can move around.  Of course, it has no idea where it is going.

Cock of the roost is an important person, who perhaps crows about himself.

To rule the roost is to be domineering.

A sitting duck is an easy target.

Picking up lame ducks is helping unfortunate people.

As bald as a coot is a synonym.  A coot is a waterbird with a white patch on its head, making it look bald.

As free as a bird means able to go anywhere.  Not a bird in a gilded cage, then!

While I was preparing this post, the set reading from The Bible Reading Fellowship notes I use was Luke 13:31-35.  There are many other references to birds in the Bible.