Lennie accidentally kills the puppy by petting or squeezing it too hard. Lennie's dream is to take care of rabbits. When the puppy dies, Lennie has no real remorse for the dog- as he's only worried about what George is going to say about the dead dog (and whether or not he'll let Lennie get rabbits). Lennie tries to justify it by saying that dogs aren't rabbits and that it was the dog's fault that it died in the first place. He even considers hiding the body.
Line 4: “Keel” has multiple meanings. The first is the part of the ship that goes along the underside that gives the ship stability in the water. This usage really doesn’t make much sense, though, because one can’t see a “steady keel” unless the boat has fallen over. The other use of keel here is as a synecdoche. The keel, a part of a boat, is used to represent the whole boat. More importantly, "the steady keel" lets us know that, despite its rough trip, the ship (like the nation) remains steadfast and true.
https://www.shmoop.com/o-captain-my-captain/ship-symbol.html