The Addams Family/WMG

The Addamses really did love Debbie.
When Debbie, and asked them whether any of them really loved her, Gomez calls for a show of hands, and all the other family members try to raise their hands. Thirty seconds into Debbie's (unbelievably selfish) Start of Darkness monologue, Grandmama seems totally on her side, and the entire family seems to sympathize with. Debbie had a hint of the family in her, but was just too selfish to see that she'd found a home and family that could love her for what she really was.

The Addamses are like that due to the ability to see ghosts
In the musical, it is notable that only the Addamses seem to notice all the undead ancestors walking around; the guests don't seem to be able to see or hear them. Logically, this means that the ability to perceive ghosts is unique to the Addamses. Presumably, prolonged exposure to the spirits of the dead from a very early age is the cause of their unique perspective on mortality and how to live one's life, as well as their odd design aesthetic sensibilities and hobbies.

Fester isn't the real Fester.
He really was Gordon Craven, and really was trying to get the Addams' money, but their insanity grew on him and he ended up brainwashing himself.
 * He may have been a bastard child.

The Addams Family are a state of mind! Or a mental disorder.
Literally, the Addams family (Gomez, Morticia, Fester, Granny, Pugsley and Wednesday) are what you become when you visit the Addams house and, instead of running away, start to empathise with them too much. It explains the slightly different versions of the family in the different adaptations. Luckily, along with a propensity towards enjoying torture and violence, you also become imbued with a supernatural toughness and regeneration that will even bring you back from the dead and will somehow 'inherit' (from a family member that never existed till then) a large and spooky manor complete with a bottomless fortune, a huge butler, a friendly severed hand and any number of family members. Of course only one such family exists in any given world, which is why all the different adaptations are all canon.
 * You Kill It, You Bought It. Kill an Addams, and slowly, you become that Addams. That is the true meaning of their family motto.
 * There's also a middle ground. The Addams Family is a state of mind, yes, and once you start to like them, marrying into the family (like Margaret Alford) comes with nigh invulnerability.

The Addams Family acts that way because they are all Nigh Invulnerable
Everyone in the family seems really hard to kill. Wednesday and Pugsley try to kill each other constantly and always live, Pubert can survive long drops and catch guillotine blades, and all the family members in the graveyard seemed to require rather brutal means of putting them down. If you and everyone in your family were that hard to kill, it's likely that you'd develop a rather blase attitude toward violence and mayhem. They don't quite realize that others are more vulnerable, so they rarely hold back when dealing with other people.

The Addams Family is in the same universe as Harry Potter
The Addams Family and their relations are magic using Wizards the same as those in the Harry Potter universe. While the Wizards of England and Europe have formed themselves into highly class-stratified societies with guilds and academies, the Wizards of North America and the United States have gone to a more independent model that we see in The Addams Family. This fits with the more casual nuclear family attitude that one sees in mainstream US culture and the fierce family loyalty that the Addams have. The Wizards of the New World would have also been highly affected by the frontier culture of the early United States: self-sufficiency, "pulling yourself up by your bootstraps". They eschew titles, nobility and cultural ranking systems, are much more casual with acquaintances and are more egalitarian and dealing with "muggles" day to day.

In "Addams Family Values," Wednesday and Pugsley really did just want to play with the baby.
It's been demonstrated that their definition of "play" involves axes and electric chairs. There's no reason for them to assume that their brother would be any less Nigh Invulnerable than the rest of the family. So, while there may be some legitimate sibling rivalry, the attempted guillotining and repeated drops from high places are just normal Addams playtime.