Why does Shakespeare use bird imagery such as "And I will make thy Swan a crow" or the talk about the lark and nightingale?
Sometimes when we have recurring motifs (like the birds in R and J), it helps to think about who is speaking because it helps us understand more about them. It also helps to think of the nature of the birds each speaker describes
For Benvolio's part, any girl can be a bird; he speaks of Rosaline as a "swan" and a "crow", neither of which are particularly flattering, and both of which reveal something of Benvolio's attitude. Swans, though gracious and beautiful in appearance are mean; crows are even worse. Benvolio tells us that she will be unattractive in comparison, but he also reveals to us that in neither instance would she make a good companion.
Similarly, Juliet's two references to birds give us insight into her nature: they are references to tamed or controlled wild things. She wants a falconer's voice to call Romeo back and she wishes her were a bird on a string that she could pull back at will.
As for the nightengale and the lark, they tell us both about Romeo's nature and also about the nature of the relationship between the two. Romeo, the romantic, is willing to let reality be named or interpreted any way Juliet wants, provided he can have his Love. This moment sparks a kind of crisis for Juliet, for she wants to control him, but she also wants him to stay safe. Just as she has with her references to the wanton's bird (she says she fears she might kill it with too much caring), she realizes that she has the potential to kill Romeo with too much control. When she finally acknowledges that it is day, she acknowledges that though she desires control, she does not want it at any cost, and the life of her loved one is more important to her than the control he gives her.
The references to the bird imagery allow us insight into the character
Because metaphors make language richer and more enjoyable. Although some of the imagery may be lost on us now, in Shakespeare's time, these references to birds would have conjured up the metaphor for most members of the audience.
For example, crows were typically harbingers of doom and thought to be quite ugly. Swans were, are still are, thought graceful and beautiful.
Your first example is from Benvolio, who is trying to get Romeo to see the fair Juliet. He boasts, "Compare her face with some that I shall show, / And I will make thee think thy swan a crow." He believes Romeo will be so taken by her beauty he will stop pining for Rosalind, who will look like an ugly crow by comparison.
Larks herald the morning and Nightingales, true to their name, sing at night. When Romeo says he has heard the lark and must leave Juliet in the morn, she does not want him to go. She tries to pretend the bird they heard was a nightingale so he will stay.
A metaphorical connection between all of the bird imagery should also be considered. When authors overwhelm a text with the same category of images (nature imagery, biblical imagery, etc.), the authors are most often using that imagery to make a broader statement than each individual use indicates.
In Romeo and Juliet, the characters are held back and held down by their situtations. Their relationships to either family keep them wrapped up in the 'ancient grudge', and the decisions of every character are inextricably influenced by that grudge. Romeo and Juliet themselves are trying to break free of these earthly concerns. They are caught up in a more divine concern, a concern of love and soul and spirit. They are trying to "fly away" from what their life has been. Bird imagery helps to reinforce this.