The scent of death, recognizable to me as any memory, wafted off of Hungry in the dry, dusty air. (1.55)
Death is a natural part of the dog's existence, and as you can see, he does not fear it. Nor does it make him sad, or seem to affect him in any way at all. It's a rather Zen approach. Or he's just stone cold.
I felt overwhelmed with a fatigue as heavy and oppressive as when I was a small puppy and my brothers and sister would lie on top of me, crushing me. That's what I was thinking about as I began to sink into a dark silent sleep—being a puppy. (4.79)
In death, our narrator returns to the beginning of life. And he does so almost literally, since he is soon reincarnated as a different puppy.
His strength was leaving him; I could feel it ebbing away. (11.36)
Bailey attempts to protect Ethan's life in a way he didn't for his own dog brother in his first life. This scene also foreshadows the end, when Bailey stays next to Ethan as he dies as an old man. Do you think Ethan will be reincarnated?
A week or so later, Smokey died. After dinner the family went into the backyard, where Ethan had excavated a big hole and they wrapped Smokey's body in a blanket and put it in the hole and covered it with dirt. […] He and Mom cried a little. (13.1)
Bailey gets to witness a burial procedure, something he doesn't get to see done to himself when he dies, even though he is later reborn. That's maybe for the best, since it would be really weird to see your own funeral.
The next day, after Mom and the boy left for school, I went out into the yard and dug Smokey back up, figuring they couldn't have meant to bury a perfectly good dead cat. (13.3)
Again, we see the dog's chill attitude toward death. It's also pretty clear that he doesn't completely understand what's going on with a burial. Death seems natural to the dog, so he doesn't understand the big deal—why not just let the cat, you know, rot somewhere in peace?
I hadn't given it any thought at all, though I supposed deep down I knew that one day I would wind up like Smokey the cat. I remembered the boy crying the day they buried Smokey in the yard, and I hoped he wouldn't cry over my death. (17.63)
The dog gives some equal opportunity ¯\_(ツ)_/¯ to death. He shrugs off even his own. However, it's a little easy for him to shrug it all off at this point, because he's already died once before. He's a pro at this, and he knows he'll probably be reborn once again. So what's to fear, really?
And again, just like that, the pain was gone—in fact, I felt like a puppy again, full of life and joy. I remembered feeling like this the first time I ever saw the boy, coming out of his house and running to me with his arms open wide. (17.75)
Once again, we see the dog come full circle in death, except this time, he isn't only thinking of himself as a puppy. He's also reflecting on Ethan's life, seeing him again as a child. Do you think similar thoughts go through Ethan's head before he dies at the end?
I anxiously lay down by Jakob's side, nuzzling his unresponsive hand. I could feel the pain working its way through his body, and the blood smell was alarming for how much of it there was. (20.44)
The dog is pretty casual about death, but this moment is traumatic because of all the violence involved. The other deaths the dog has witnessed have been natural ones, but this one is caused by a gunshot. Don't worry, Jakob ends up surviving—but at this point, the dog doesn't realize it yet, and it's scary for her.
I wondered if there were anything I would have wanted to do one more time—Find? Swim in the ocean? Stick my head out the car window? These were all wonderful things—I had done them all, though, and that was enough. (25.73)
In her final moments, Ellie appears to be evaluating her doggie bucket list to see if she had done everything in life she wanted to do. She did, but then again, she had simple desires. How does her evaluation of her own life moments before her death compare to what humans do when they are facing their own mortality?
There was no pain, no fear, nothing but the sense that my brave boy was going where he was supposed to go. (32.45)
The dog is confident that Ethan is going where he is "supposed to go," but he doesn't say exactly where that is. Does the dog believe that Ethan will move on to an afterlife? Why doesn't he think that Ethan will be reincarnated as the dog himself was? Or does he maybe believe that?