All of the movie's ridiculous plot points and such can be easily explained when you remember that the story's being told from the perspective of a 7-year old with an overactive imagination.
Tim's story could be seen as a surreal allegory of how he dealt with the news of his parents having a baby during those 9 months. He was afraid the baby would be more successful than him, or take away his parents' attention from him. The rest of the story could be creative ways he went through the Five Stages of Grief.
Denial: His parents being oblivious to how smart Boss Baby is.
Anger: Tim's dislike of Boss Baby which comes to a head during their argument.
Bargaining: Boss Baby bargaining with Tim that he won't hurt Lam-Lam so long as he keeps the babies' operation a secret.
Acceptance: When Tim writes a letter asking Boss Baby to be a part of his family.
Besides entertaining his daughter, there's another reason why Tim is telling this story— it occupies his daughter while her mother is giving birth.
Towards the beginning of the film, Tim tries to catch the Boss Baby talking by using a portable cassette recorder to record everything he and the other babies are saying. It might be strange that Tim is using such an old piece of technology, but then it's revealed at the end that the film actually takes place in the 2010s (the film even very deliberately shows that his child is born on 3/31/17), placing Tim's childhood roughly in the late 70s or early 80s, when cassette tapes were extremely common.
The film being a vaguely 70s-80s period piece fits extremely well with the plot of the movie, as it means the sharp decline in interest in babies that Boss Baby is trying to look into would be the post Baby Boom birthrate decline.
It initially seemed like a Plot Hole that Francis Francis remembers being Super Colossal Big Fat Boss Baby despite the babies losing their memories when they get older, but then you remember: Boss Baby said it was specifically losing their pacifier that causes the memories to be taken away, and he still has it as an adult. He must've somehow hidden it from both Baby Corp when he got fired, and then his adoptive parents after that, preventing him from losing his memories!
The properties of the pacifiers themselves may or not be a reference to Fionn Mac Cumhaill, who was cooking the Salmon of Knowledge when he burnt his thumb on the grease. He stuck it in his mouth to cool it and gained all wisdom from the Salmon. Now he can access the wisdom, but only when he sucks on his thumb. Sound familiar?
Given what happened to the tape, there's a good chance that Tim would have gotten Boss Baby killed by catapulting him out the window. No wonder his parents ground him so quickly.
Francis E. Francis wants to destroy Baby Corp because the formula stopped working for him, due to him being lactose intolerant, which gets him fired. Francis has been stopped, but if there's another baby that goes through the same thing as Francis, then Baby Corp will be back to square one.
How did Tim learn about sex at seven years old (or even younger)?
Some parents give the talk a bit early. Not nitty gritty details, but explaining the basic biology of reproduction to a seven year old is not unusual.
Especially if the parents in question are expecting.
In the Forever Puppy infomercial, it says that Puppy Co. is not responsible for the eventual extinction of babies. Puppies that stay puppies forever vs mortal humans? Uhoh.
