The tests James and the soldiers/sailors/marines/etcetera are doing shortly before James is recruited as J had different reasons for being used than those that are apparent:
The first one, where they have to fill those documents, is actually about being able to think outside the box and perhaps even a willingness to break assumed rules. They were sitting down on chairs that, while comfortable, had no place to allow the candidates to write the answers. James decides to break the assumed rule of "do not touch the table" and to think out of the box, by grabbing the table and moving it so that he can write without having to twist himself.
The second one, the shooting range, is about both accuracy, critical thinking and tolerance. All other candidates shoot indiscriminately against all the aliens, figuring them to be hostile just because of how they look like, but James does not just look at the aliens, but he also sees what they are actually doing, which are inoffensive activities, while, when he looks at the little girl, he is able to see that she is quite out of place due to what she is carrying (those quantum physics books) and the place where she is (in the middle of the 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.
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 first movie, it's specified that the number of aliens is quite low- ten thousand on Earth, mostly in Manhattan. But then, in the second, J goes to the middle of relative nowhere, Massachusetts, to find K, and there's randomly a bunch of aliens in the back room. Why would there be so many aliens in some random post office when we know there's so few in general? Because they wanted to work with K, even if he had no idea who (or what) they were. It's a moment of fridge brilliance that turns into a crowning moment of heartwarming.
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 third movie, Andy Warhol is secretly an MIB agent. His Code Name is Agent W. The problem is that the naming convention for agents is the first letter (or first two letters, presumably, for Agent AA) of their first name. So Andy Warhol isn't even his real name! It's just another pseudonym.
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.
Men In Black, the neuralyzers in particular. 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.
The organization exists basically to prevent whatever the aliens and its technology might inflict (panic, misusage)... so they must realize such a thing. K explicitly states in the first film that the neuralyzer alien tech from "friends from out of town".
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.
In the second Men In Black movie, while Agent K is still regaining his memory, at one point he shoots off the head of an allied alien. Agent J asks him if he regained his memory, since he appears to know the alien's head would grow back. "His head grows back?" This is played for laughs, but when looked at logically is rather shocking: K is willing to kill an ally merely because he was annoying?
Though it is possible that K knew he could get away with it, but not WHY he could. Or maybe he knew all along and had a laugh at J's expense. It wouldn't be outside K's character.
He was also probably disoriented from being re-neuralized(?).
Muscle-memory, perhaps? Given how the 'interrogation' went in the first movie, this almost seems like a ritual between them.
Why are K and Zed suddenly able to jump six feet in the air and hover in the second movie?