The Belchers have excellent family harmony despite their diverse and clashing personalities. There are many other reasons as well for their wonderful dynamic, but it's interesting to note that there is at least one trait common to every member of the family: creativity. Linda and Gene are both musically inclined and actually quite good. Linda is very good at ad-libbing lyrics to songs along with melodies within a moment's notice, while Gene has genuine musical talent (seriously, a 4th-grader writing "Electric Love" probably qualifies as a genius). Tina is of course a devoted writer, probably stemming from her elaborate fantasy life, and Louise is a master planner who is willing to think outside the box for solutions to problems (even if outside the box means "verging on psychotic"). Finally, Bob is extremely creative in his cooking. The show's producers have said that the Burger of the Day is not merely Bob playing with puns, but that it shows how much thought and work he puts into his creations. He daily sits down and dreams up a logical, interesting variation on the same basic concept, and never lets himself get lazy.
In the first season episode "Burger Wars," Mr. Fischoeder runs a raccoon sanctuary in the building next door to Bob's Burgers. Over the course of the series Little King Trashmouth, El Diablo, and a number of other raccoons living in the alley behind the Belchers' become part of the narrative, meaning either the sanctuary was shut down and the raccoons now roam the neighborhood or the sanctuary's back door needs tighter security.
In the second season episode "Moody Foodie," the food critic visits the restaurant disguised as a Hasidic Jew. Bob freaks out about his napkin as a "tell," and it's played as if it could just be a random customer, not the Moody Foodie. But there's an easier tell for the audience - he orders a cheeseburger. That's about as non-kosher as you can get.
Another layer to Tina's anger at Mr. Frond's cheating on Aunt Gayle is a justifiable reaction...especially since she was the one to convince Louise to stop their aunt from dating their school counselor. At that moment, Tina was angry at not just the act, but the fact she stood up for him.
The episode "Spaghetti Western And Meatballs" has Mr. Frond introduce his A.B.S. method of conflict resolution, as well as a subplot that climaxes with Gene and Bob being confronted and pursued by a shirtless Choo-Choo and his father. It's something of a Stealth Pun that Choo-Choo and his dad are therefor literally using their "abs" in a conflict. Based on the events of this episode, Choo-Choo apparently always takes his shirt off when he's ready to start a fight: Maybe Mr. Frond saw him do this on the school grounds once before and the whole "A.B.S." acronym took off from there.
Why did Bob name his son Eugene/"Gene" instead of Robert III (aside from Eugene also being the name of Gene's voice actor)? It's because of the pressure Bob got from being the Jr. of a man who has to make sure everything is done exactly the way he wants it. He didn't want to risk Gene feeling he needed to be another "Bob"!
In "Art Crawl", Louise becomes a drill-sergeant-like manager to no one's surprise. As the episode goes on, she is overseeing Andy, Ollie, and another kid with red hair who is unrelated. Among all of the shouting that she hurls, Louise seems to switch gears for a minute when talking to Andy and Ollie at the table.
Louise: (Sweet, loving voice) Andy, Ollie, come 'ere...(puts hands on both and pulls them in) Listen, I'm pushing you because you've got talent, babies. And I'm gonna take care of you, oh, you better believe it. But first you gotta make some money for Mommy. OK, babies?
This line isn't particularly meaningful until the viewer looks at the context involved. It's made clear in the show that Ollie and Andy are raised by their father (and not necessarily in a loving way). Their real mother is nowhere and is never mentioned...a fact that Louise surely knows. What Louise could possibly be trying to do with this line is intentionally invoke Ollie and Andy's emptiness for the love of a mother in their lives, and use that deep psychological emptiness to get better art out of them. And worst of all? Louise abandons them there. And at the end of the episode, after Louise has left the table for several hours, Ollie and Andy are still waiting for their "Mommy" to return. The red head child has left, but Andy asks Ollie, "Do you think she's coming back?" Granted, Louise does eventually return (with the express intention of collecting more paintings from them!) but it seems to be a pretty cleverly conniving tactic on her part.
Interesting note, adding more to the Fridge Horror, it was implied in "The Belchies" that their parents are divorced (i.e he tells them, "I thought you were at your mother's.") and its very clear that said parents' divorce has affected them greatly and they rarely get to see her, along with the fact that said divorce might have occurred while they were very young. In which case, it makes the above even more depressing.
It's pretty clear that the Pestos' family problems have left the two with serious abandonment issues. The two are constantly together and become terrified to the point of a full emotional breakdown at the thought of being separated.
What is Louise going to be like as an adult? There's no indication that her Comedic Sociopathy is a passing phase, or that it's just an act covering up a harmless little girl. She's actually pretty friggin' dangerous, even at nine years old. She nearly cuts off her brother's ear, causes an angry mob to descend on her dad's shop because it amuses her, and tortures a grown man (psychologically and physically) to the breaking point for occupying her room. What's going to happen when she's physically and intellectually capable of doing more damage?
The fact that this has been implied to be an act may just make it worse as she has no reason to behave this way and we don't even have any real confirmation if it is an act or not.
It's actually becoming more and more likely as the show goes on that this is just a phase. Present-Louise is a much more thoughtful and caring person than past-Louise. Can you imagine Louise from season 1 doing things like kissing Regular-Sized Rudy, or stepping in to stop Pocket-Sized Rudy's bullies, or working hard to save a doll from being destroyed and then bringing it home? As for why it's an act, it's implied that the Belchers don't monitor her TV shows very closely, since she's apparently seen Game of Thrones.
The family's mental health can be scary once you think about it. Bob has a severely addictive personality to drugs and alcohol, Linda and Gayle have very alarming personality problems (callousness, egotism, Munchausen syndrome, lack of empathy, narcissism, violent tendencies), Louise's above-mentioned sociopathy, Gene's ADHD and Tina's possible autism, along with that the fact that the family's poverty means that they can never get any therapy or medication that they truly need, is a very sad and terrifying thought, as well as a common reality in the United States.
In the season 4 finale, Felix Fischoeder holds Bob and his brother Calvin at gunpoint, and ties them up under the pier with the intention of letting them drown. While Bob is reasonably afraid, Calvin, whose eye was cut out by Felix, reacts with nothing but amusement, believing Felix not to be tough enough to go through with it, even mocking him when he does return. How many times did that happen that he doesn't care about his brother trying to kill him?
Nurse Liz from "Lice Things are Lice". We find out that a regular hospital/clinic wouldn't take her, so, knowing that and after seeing what she does in that episode, one has to be wonder as to how she even got employed at the school.