The entrance tests James and the soldiers/sailors/marines/etc. were given had different reasons for being used than those that are apparent:
The first one, where they have to fill documents, is being able to think outside the box and perhaps even a willingness to break assumed rules. The candidates were sitting down on chairs that, while comfortable, provided no place to write down answers. James broke the assumed rule of "do not touch the table" and thought out of the box, grabbing the table and moving it so that he can write without having to twist himself.
The second one, the shooting range:
This test is about accuracy, critical thinking and tolerance. All other candidates shoot indiscriminately at the alien targets, judging them as hostile just from their looks. James didn't do that; he saw what they were doing, which are inoffensive activities, while, when he looks at the little girl, he's able to spot out she's out of place due to what she is carrying (quantum physics books) and the place where she is (in the middle of a ghetto, surrounded by apparently threatening aliens, without looking scared), which are not normal qualities in a little 8-year-old human girl. Not to mention, that, since there are many shapeshifting aliens, any human that behaves out of character is a serious candidate to being an alien.
Another possibility, since James seemed confused by the "aliens" setup and fired a single blind shot, is that he didn't mean to hit the little girl, who was simply included as an "innocent" target you weren't supposed to hit. However, he redeemed himself by coming up with a totally spontaneous plausible-sounding lie, an essential skill of an Mi B.
This is capped off by Zed's line after the shooting test: "Congratulations, gentlemen. You are everything we've come to expect after years of government training." In this case, a willingness to take things at face-value and do what's expected of them, whereas Edwards is willing to question his assumptions. A nice, subtle little "up yours" from Zed at the soldier-boys.
This extends even before the official tests. The question of whether they know why they're being recruited is a way of seeing whether they're even willing to admit when they don't know anything or if they'll simply fumble around trying to look special.
It's entirely possible that the other 'candidates' weren't candidates at all, but simply part of the show to evaluate James. As a police detective, James has exactly the skill set the MiB are looking for, and has already displayed at least some of the tenacity an MiB agent needs for the job. The whole 'test' was for show; they knew James was their man and just needed to confirm it (by evaluating his ability to assess a situation rapidly, think outside the box, and not be intimidated—useful skills for both a police officer and a Man In Black).
In the third movie, the scene about J's dad at Wu's restaurant may have been Played for Laughs or something else. It becomes harsher when we find out the identity of J's father and his fate. It also manages to kind of explain K's role as a father figure to J.
In the first two films, the majority of the aliens were done via CG with some practical tossed in. In the third film during the 1969 scenes, most of the aliens are done via practical makeup/animatronic effects via Rick Baker, to give them a more "retro" feel similar to how they would have looked during that era.
In the third film, Jay points out that the rocket packs, even those designed by secret agencies that may possibly have hints of extraterrestrial technology powering them, were Awesome but Impractical and there were plenty of reasons why they don't use them in the modern day. He then mentions the infamous red button feature that comes standard on all MIB vehicles by then, which weren't around yet. The scientists and technicians overhearing him at this point in time, might've been what prompted the development of the hidden car rocket systems activated by the red button in the first place.
In the first film, Jay mentions having suspected that his 3rd grade teacher was an alien from Venus, to which Kay confirms was from Jupiter, or rather, one of its moons. How Kay could possibly know about Jay's childhood becomes more understandable after the third film, in which Kay sees himself as a father figure after accidentally getting Jay's father killed in 1969. Plus, it shows that Kay always thought Jay had the potential to become an MIB agent.
In the first film, as Jay and Kay speak to Beatrice, she offers lemonade to them, but Jay grimaces while drinking it, not-so-subtly spitting it back into the glass. But the lemonade's bad because the Edgar bug ate all the sugar.
Going from killing witnesses in the comics to neuralizing them in the movies doesn't make sense just because of Adaptational Heroism. Eventually someone in the comics would notice all those people disappearing.
Assuming they can wipe out entire memories, someone's entire personality, life decisions, changes, thoughts, plans could be obliterated and the victim would essentially be a blank slate who could be manipulated by the user. Now imagine if someone less honorable got their hands on the devices. It may be why they're so aggressive with it (and other tech) in the first place in terms of using it (flashie thingie) or keeping it a secret. Allowing anyone not MIB (and limiting how many field agents have one) to even having an inkling of the sort of technological advantage the organization has could be very very very dangerous.
You will never remember your experiences in MIB if you live long enough to retire. All those times you had, and all those agents you meet will be like they never happened. Could explain why K and the other older agents are so closed off.
The Neuralyzer is straight out of the comic, though there it is more of a general purpose hypnosis beam. It's used for questionings and memory erasure...and by Agent K to brainwash a middle schooler into going on a shooting spree a la Charles Whitman style when he hits 19.
When J neuralyses the video shop owner and tells him to move out of his mom's house, the guy picks up a shovel. Funny, maybe, but just thinking about it, was that guy planning to kill his mother?
In the third movie, it's stated that J's memory wasn't changed by Boris's time changing because he was present in the event. We later discover that J was present AS A CHILD, and son of the colonel overseeing Apollo launching. Which means that in the Boris Wins timeline, the young child probably witnessed K's death, and likely his father's death as well(And maybe seen Boris' real form), before getting mind-wiped by the MIB clean-up crew. And the mind-wipe as a child would explain why J wouldn't remember the event in ANY alternative timelines the time travel change created, but would still be aware of it.