In the pilot of the Disney series, Doug remembers having fun with Mr. Bone the assistant principal and Skeeter informs him Mr. Bone was a dick. There was an episode of the original series where Mr. Bone lightens up after having surgery and acts like a kid again. He also befriends Doug and this could've been a flashback to that.
How does Mr. Dink afford the latest and "Very Expensive" tools and gadgets? Early in the series he mentions that he's a writer, he's probably testing and writing product reviews for them.
It's also been stated that he and his wife have no kids and two incomes so naturally there's a lot of extra money lying around. This adds further Fridge Brilliance if you consider "DINK" is also an acronym for "Double Income, No Kids".
In the episode "Doug Clobbers Patti" he tries to break a winning streak he's having against Patti Mayonnaise by breaking mirrors, running under ladders, and doing this weird ritual Skeeter's aunt has. Afterwards, he's tripping over thin air, his bike tire blows for no reason and a car splashes mud all over him. When he starts a bowling match against her, he's still winning, without even trying. Why? He wants to lose, and his bad luck is keeping it from happening!
Judy is normally a bit of an antagonist towards Doug, but has a lot of Pet the Dog moments (moreso than Roger). This makes sense if you do have a sibling, actually - Doug is an Unreliable Narrator so he'll up Judy's Jerk Ass-ishness a bit. Plus, Judy's just getting older - usually most siblings get better to their Annoying Younger Sibling when they get older and more mature (keyword being usually...tropers can name a few siblings they hate well into adulthood).
The pilot to the Nickelodeon series foreshadows the plot of Doug's First Movie, since it involves searching for a rare creature in the lake.
Patti is a good singer in the Nickelodeon show, but is Hollywood Tone-Deaf in the Disney version. It's possible this is due to her going through puberty.
Why do the rest of Doug's relatives have different skin tones than him? Judy's tan, his dad's orange, his mom's pink, and his Grandma is yellow? He wasn't adopted and the rest of his friends have the same skin tones as their parents.
This could be chalked up to Doug's Unreliable Narrator status, given how everyone else is a different color scheme due to being a cast of colorful characters compared to (the admittedly dull) focus protagonist who has a realistic skin tone. Not to mention how the colors could very well reflect how he perceives his family members. A bit of a stretch, but it could account for his Mother being pink (being the sweet but sometimes embarrassing maternal figure) and Judy being tan (older age and perhaps also her stage lighting-tanned melodramatic actress persona).
If Beebee Bluff's family is so rich, why does she go to a public school?
Because her family is the only one rich enough to pay for private school and Bluffinton doesn't have one. Or she just wants to go to public school.
In the episode "Doug's Career Anxiety" Doug is going the task of being in charge of a school assignment with Connie, Beebee, Chalky, and Skeeter. When things get hard to handle he runs off and has a fantasy where he's been running away everyday of his life. Connie is nowhere to be seen in his fantasy. Did Doug imagine that Connie died? Does Doug want that to happen to her?
In the episode "Doug's Older Woman" from the Disney series, there was the girlfriend dujour that pointed out how funny Doug was acting multiple times, with him denying it (since he doesn't remember the fantasies he was acting out). Now since she only appeared in that episode she probably didn't witness some of his worst fantasies (like the fantasy he has while in the middle of traffic), but the fact that she is named Cassandra does add some Fridge Brilliance on top of the Fridge Horror.
Unreliable Narrator mean anything to you? We're probably just seeing things colored by Doug's point of view. When he's gotten hurt as part of the actual plot, his friends react appropriately.