When Freddy discovers Jason's weakness to water, he adds insult to injury by showing the dismembered head of Jason's Mother in front of his child form. Now Jason is a Genius Bruiser when it comes to killing, everything else however he is not so smart. So when he sees his mother's head he comes to the logical conclusion (in his point of view) that Freddy killed Mrs. Voorhees. When Freddy enters the real world, Jason even stops going after the others just to confront Freddy.
Jason's being too afraid to go through water initially seems like a continuity error, as severalpastFridaymovies have repeatedly shown him having no problem wading through water to go after a target. Then you remember this happens when he's in his own nightmare, and even though he's been able to overcome it in the past, he's become much more susceptible to his fear of it in the dream world.
It could also be justified by looking at how things ended up for him the last few times he was in the water. In Part VI and VII, he was trapped at the bottom of Crystal Lake, and in Part VIII he wound up being separated from his home.
Freddy's strange bout of Offscreen Teleportation could be explained by him running like hell and scampering up the platform while Jason is taking his time to stand back up after the torpedo strike.
Kia's use of the word "faggot" to offend Freddy. It was still somewhat of an insult in 2003. But Freddy died in the 60s. Kia knew that he came from a time when being accused of homosexuality was much more serious.
Jason starts attacking when Freddy is about to rape Lori in her nightmare - just as he killed the raver about to rape Gibb. He also seemingly comes out on top in his fight with Freddy. Since this is partially a Friday the 13th movie, the old rule applies - Jason kills the horny and the Asshole Victims before targeting the more chaste and innocent. So Jason apparently gets the kill on the evillyperverted Freddy ahead of the comparatively far more virtuous teens and thus fulfills his movies' tropes, but since the A Nightmare on Elm Street movies usually end with the implication that Freddy is Not Quite Dead, Freddy is suggested to still be alive and thus fulfills his movies' tropes.
Rather a subtle one at the start with Freddy. Krueger is in Hell, and (theoretically) suffering the torments of the damned. What do we see of his 'torment'? Freddy surrounded by pictures of all the children he murdered, while being unable to kill any more. That is Freddy Krueger's Hell. Knowing that he killed dozens of children... And can't kill more of them. He's THAT evil.
Jason's version of Hell is in Camp Crystal Lake. He's killing teenagers. They're all very bad. But no matter how many he kills, they won't die. They keep coming back and he can't finish the job.
It is heavily implied that Freddy killed Jason to begin with. If Freddy was murdered by the parents of Elm Street before their children were born...then that would make the date of Freddy's death at somewhere around 1970. Jason died in 1957. In other words...Freddy murdered children for at least 13 years.
Well, Jason's nightmare implies that his drowning wasn't wholly accidental, and that other kids bullying him pushed him into the water. . .
We see (in Freddy Nightmare-O-Vision, granted) that Pamela Voorhees wasn't entirely wrong to blame the counselors, if this version of events is remotely true. While the other kids were bullying Jason, forcing him into the water where he couldn't swim strongly enough to save his own life, the counselors were doing everything but keeping an eye on the kids. If they'd actually been paying attention, they would have noticed a childish prank about to go lethally awry.