What's up with the puppy? Why do you suppose Fitzgerald included it as an element in this chapter?

Myrtle decides en route to the apartment Tom keeps for her in Manhattan that she wants Tom to buy her a puppy. He indifferently complies.

The puppy represents Myrtle. Myrtle might want a protective "police dog," but what she gets is a tiny puppy that "cowered" and is, as the racist and classist Tom thinks of people like Myrtle, of "an indeterminate breed." The vendor tells Myrtle that she is getting a male, but Tom...

Myrtle decides en route to the apartment Tom keeps for her in Manhattan that she wants Tom to buy her a puppy. He indifferently complies.

The puppy represents Myrtle. Myrtle might want a protective "police dog," but what she gets is a tiny puppy that "cowered" and is, as the racist and classist Tom thinks of people like Myrtle, of "an indeterminate breed." The vendor tells Myrtle that she is getting a male, but Tom rather brutally informs her that (like her) it is a "bitch." 

Like Myrtle, the puppy is out of place. Myrtle might put on airs, but she has no more real place in Tom's world than the puppy. The "elevator" boy has to find a box of straw and milk for the puppy, to which he adds a tin of dog biscuits. One of the biscuits "decomposed apathetically in the saucer of milk all afternoon": a sad and unpleasant image. Myrtle wants to buy it a "little collar," a symbol of ownership as much as the chiffon dress from Tom she flounces in. Despite her bravado and strong personality, she is, like the puppy, very vulnerable, as we see when Tom hits her. Like the puppy, she is one of Tom's "purchases." 

That the puppy symbolizes her is emphasized near the end of the novel, when Tom tells of crying when he sees "that damn box of dog biscuits sitting there on the sideboard." He also says that Gatsby ran over Myrtle "like you'd run over a dog."