Corrected entry: The opening of the film says it takes place in July 2002, but when we first see Will Smith you can see the World Trade Center in the background which had been destroyed ten months before. [The film was made before the 9/11 attacks, and the towers were left in as a sign of respect - still a mistake, but there's the reason.] (00:03:25 - 00:06:45)
Correction: To add to the other correction about the "XYZ" buildings, there's another visual point of reference to their location in the City. When Will is thrown into the air and lands on the produce, across the street behind him is the 6th Ave entrance to the Comcast building (where Fallon tapes the Tonight Show for example), which is up the block from Radio City Music Hall, at Rockefeller Center in Midtown Manhattan. The World Trade Center complex is (and was) located in Lower Manhattan, by the West Side Highway.
Correction: Not only is that not the WTC, they specifically reshot and edited scenes to remove the twin towers. The exact opposite of your claim that they left them in as a "sign of respect."
Corrected entry: When Agent Jay is in the yellow tent with Serleena's tiny ship, he pulls out his video communicator and begins to talk to Zed. Zed can see Agent Jay via the communicator but when Jay closes the communicator and puts it away, Zed still has a view of Jay speaking with him over the big, egg shaped monitor. How could Zed see Agent Jay over the monitor if there was no video device available to transmit the conversation? (00:24:05)
Correction: The device Agent Jay uses allows him to see HQ, not the other way around. In one shot of Jay on the monitor, you can see part of the spaceship, meaning the image of Jay did not come from his monitor, but some other device monitoring either the spaceship or Agent Jay. When Jay closes his visual communicator, the monitoring device is still active and thus HQ can still see him.
Deliberate mistake: When MIB agents exit the lift to get into the main hall of the MIB Office, the lift opens halfway up the wall and they need to go down a little platform. But when K and J return during the lockdown and face the trashcan with guns, the lift opens at floor level, so it can be conveniently peppered with bullets. (01:02:05)
Suggested correction: The building is in lockdown so it is no surprise that the lift is changed to a less flamboyant version. Even in the first scene, there are some doors there.
Corrected entry: When K shows his video store card to the clerk at the video store, the clerk remarks that the card hasn't been used since before she was born. How would she know that fact if she hadn't been born yet when it was used? Perhaps the card was only discontinued right before she began working there.
Correction: First, it was probably just an exaggeration on how old the card is. But why wouldn't she know about it, she's obviously aware of the card, so she could have been told about it or come across one by another long time customer.
Continuity mistake: When they land on the roof of the launch site, Jay gets out of the passenger side, even though he and Kay switched spots after which Jay was driving/flying.
Corrected entry: When K and J are in Times Square after being flushed, they are covered with droplets of clear water. However the water on the walls of the cylinders as well as what they were in was blue. (00:35:35)
Correction: There's no mistake, they just appear clear, but it's all the same blue liquid. In fact, in one shot you see Jay spit out the blue liquid and even as the blue liquid is seen, some of the drops on his lips and chin appear clear.
Given the droplets were the same size on the walls of the pod as they were on J and K, they would not appear to be blue on one surface and clear on another. If the drops were larger in the pod vs on them, then I'd agree that the color could possibly be blue vs. Clear - but they weren't so they should be blue regardless of being on a person vs. Pod wall.
Correction: There's a lot more to just droplet size that affects what color you see. But the point being, he is seen spitting out blue liquid and that same liquid appears clear (as opposed to seeing him spit out blue liquid, have shot change, then another shot back to J where the liquid is now clear).
Correction: That isn't the world trade center, Those are the XYZ buildings in New York City, a part of Rockefeller Center.