Jack. Is he acting like a 13-year-old would during the apocalypse, albeit trying to put his own spin on survival, or did he have a complete mental break and is hiding it with his video game terminology? He's making the best of a terrible situation, but some things he does makes it seem he does not fully grasp the reality of what's going on. Multiple moments involve him endangering his friends that, while not for selfish reasons, put them at risk. Also, he views things through a video game lens (like his friends' "happiness meter"), and that makes it seem like he is not entirely sane.
Despite what the title would suggest, the show is a comedy for the most part, and Jack and his friends aren't as traumatized like one would expect thirteen-year-olds to be in dire circumstances. They also don't seem too bothered about the fact that people they know (Quint's babysitter, various teachers from their school) have become zombies, even though there's no cure for it, and zombies getting killed is a plot point in Book 2.
Jack, however, does reflect on this a lot, most notably in his conversation with June. He is aware of how horrible things are, but he chooses to make the best out of his situation and not think about it too much.
While trying to siphon gasoline, Jack comes across an abandoned car with a family vacation picture inside it. He comments he wishes he had a family and is saddened to think what might have happened to this one.
Angst is recognized over the course of Book 2 and is a main plot point in episode 7 when the kids visit June's home.
Author's Saving Throw: The book's weak point is depicting June as a slightly generic at best The Smurfette Principle. Somewhat justified, as the book itself is written from Jack's point of view. The show fixes the aforementioned issue by adding in some scenes from June's perspective.
Ho Yay: Dirk and Quint get a little bit of this. Most notably that Dirk warms up to Quint very fast, especially after seeing all of the tech Quint makes.
Unexpected Character: The video game based on the show, The Last Kids on Earth and the Staff of Doom, features Meathook as one of the bosses. While Meathook was introduced in the last scene of the show's Book 3, he is more prominent in the following book, which was never adapted for the show. Confusingly, Jack and the others treat Meathook like they already faced him.
What Do You Mean, It's for Kids?: In spite of being aimed at children, the show itself does actually make the monsters more threatening to the point of almost always endangering our main characters (who are in fact, 13-year-olds). Even the zombies are Nightmare Fuel for children who have phobias of zombies. There is also a graphic depiction of gore in some scenes (first, when Jack stabs Blarg in the eye, and second, when he kills Blarg by stabbing its brain).
Made even worse for the original books and even the show, where it's gotten darker ever since Book 2 ("Zombie Parade," in particular).
The novel series by Max Brallier
Complete Monster: Thrull is a power-hungry sociopath with a serious case of Chronic Backstabbing Disorder. Thrull gains the trust of hero Jack Sullivan and his friends to trick them into completing a magical bestiary, with which he plans to summon Rezzoch, and feed the monsters he befriended to his master as an appetizer; Thrull does this despite knowing that Rezzoch could destroy humanity. Forming an alliance with Evie Snark after his plan is thwarted, Thrull betrays her as well and murders the heroic Bardle for more power. Thrull proceeds to use his newfound power to raise an army of the dead to help summon Rezzoch. When Thrull locates the heroes in a mall populated by thousands of benevolent monsters, he orders his army to massacre everyone when they refuse to surrender. When he has Jack at his mercy, Thrull gloats that he will resurrect him as a mindless slave after killing him and use him to kill his friends.