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.
Speaking of Wu's restaurant, K flying into a No-Holds-Barred Beatdown on him after discovering a Spikey Bulba is part of the menu makes more sense of it triggering a traumatic memory of Boris considering he loves eating Spikey Bulba and suspected he must've escaped and is nearby, rather than it being Hand Waved as an Out-of-Character Moment.
"Have you ever flashy thing'd me before?" in the first movie is now far more poignant, almost to the point of being Fridge Horror.
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.
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 third 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.
While K's increased stoicism in this film can be seen as Flanderization, especially in comparison to the first film, a lot of events between the two films affected his persona. In the first film, there was the hope of retirement in the form of being neuralized and reunited with his old girlfriend/wife. (He would be relieved in not knowing all those 100+ memories he didn't want.) By the second film, he is emotionally detached from civilian life to the point in which his wife left him. He is resigned to the fact that he will always be a MIB agent until he dies whether he likes it or not. Also, by the end of the second film, he had to relive the death of Lauranna and say goodbye to (presumably) his daughter. No wonder he doesn't crack a smile anymore.
Lunarmax is shown to have personnel that are largely equipped the same way as conventional prison guards, complete with an armory mostly filled with ordinary guns, in sharp contrast to the fancy energy weapons used by MIB agents. While this could be handwaved as disguising the reveal that it's on the Moon, conventional firearms would be far less likely to cause destructive decompression as opposed to said laser guns, which Boris demonstrates during his escape.
The Men in Black have always used large Ford sedans, but it’s especially fitting that in the sixties its car of choice was called the Galaxie.
Zed's year of birth being 19XX. Given the Men in Black's secretive nature (he and K worked together for 40 years without the latter learning anything about the former's personal life), the only people who would know the exact year are those who recruited him and, if any of them outlived him, they would have kept it a secret.
Boris not only killed his "girlfriend" and a bunch of guards during his escape, but probably killed a lot of other prisoners, as many walls and airtight doors as he blasted through. Fortunately, all of this is mitigated by the reset after Boris is properly defeated.
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.
