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 70s. Kia knew that he came from a time when being accused of homosexuality was much more serious.
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.