[He] walked heavily, dragging his feet a little, way a bear drags his paws. His arms did not swing at his sides, but hung loosely. (1.4)
Should we keep count? This is instance #1 of Lennie being compared to an animal—a bear, no less: a massive, occasionally violent creature.
Lennie dabbled his big paw in the water and wiggled his fingers so the water arose in little splashes; rings widened across the pool o the other side and came back again. Lennie watched them go. "Look, George. Look what I done." (1.9)
Lennie doesn't get hands—he gets "paws," and he's fascinated with how those paws can affect the natural world. It's also as though, like an animal, he doesn't quite understand cause-and-effect.
"What you want of a dead mouse, anyways?"
"I could pet it with my thumb while we walked along," said Lennie. (1.36-37)
First, gross. Second, Lennie doesn't seem to have the same hang-ups about death as other people. His mental disability makes him closer to an animal than to a human—which makes us think that Steinbeck is saying that the difference between men and beasts is more of a continuum than a sharp divide.
The old man came slowly into the room. He had his broom in his hand. And at his heels there walked a dragfooted sheep dog, gray of muzzle, and with pale, blind old eyes. The dog struggled lamely to the side of the room and lay down, grunting softly to himself and licking his grizzled, moth-eaten coat. The swamper watched him until he was settled. "I wasn't listenin'. I was jus' standin' in he shad a minute scratchin' my dog." (2.65)
Candy's relationship with his dog is a lot like George's relationship with Lennie: they both care for things that other people can't appreciate.
"She slang her pups last night," said Slim. "Nine of ‘em. I drowned four of ‘em right off. She couldn’t feed that many." (2.186)
Slim doesn't sentimentalize the natural world. He knows that the dog can't nurse nine puppies, so he kills five of them to save the others. There's no moral lesson here, unless it's that a man's "gotta," sometimes.
Lennie covered his face with huge paws and bleated with terror. He cried, "Make ‘um stop, George." (3.248)
Here's Lennie with his "paws" again, and this time he's "bleating," like a lamb about to be slaughtered—which, in one reading, is exactly what happens. He's an innocent who gets killed to please the men in charge.
[Candy] said miserably, "You seen what they done to my dog tonight? They says he wasn't no good to himself nor nobody else. When they can me here I wisht somebody'd shoot me. But they won't do nothing like that. I won't have no place to go, an' I can't get no more jobs." (3.222)
Here's the thing: the world treats old men like Candy just as badly as they treat old dogs, and maybe even worse. They'll kill the old dogs, but they'll make the old men suffer.
His huge companion dropped his blankets and flung himself down and drank from the surface of the green pool; drank with long gulps, snorting into the water like a horse. The small man stepped nervously beside him. (1.5)
Instance #2 of Lennie being compared to an animal—this time, a horse who has to be kept from drinking too much water. There's a sense here, at least, that man has some responsibility to control the natural world.