Battle Royale/WMG

Battle Royale and Nineteen Eighty Four are in the same world
Obviously, the totalitarian Republic of Greater East Asia is the same nation as Eastasia. Orwell says that Oceania, Eurasia and Eastasia are all the same, and the similarities are striking. Battle Royale shows the rewriting of history books (Noriko is genuinely surprised that people had it better in the "feudal times"), publicly broadcast executions as entertainment, propaganda against the "evil imperialists", and even a Big Brother figure in the Great Dictator.
 * Supported by the fact that, if 1984 is to be believed, Eurasia and Eastasia never ally with each other, which is unlikely if Eastasia were China, but makes sense if Eastasia is Imperial Japan. After all, after World War II, neither Russia nor Japan liked the America or the rest of the Anglosphere, but they both respected them for being Badass warriors. If the war ended in a conditional armistice, rather than nuclear weapon-induced unconditional surrender, Japan would have retained considerable territory on the Mainland and the world would not have developed a healthy aversion to the use of nuclear weapons.

Battle Royale and Children of Men are in the same continuity
BR is set a few years before Children of Men, before everyone outside Britain went nuts, but the first symptoms are already showing. The unruly schoolchildren are the last generation -- and completely detached. Taking parts of BR 2 (and only some parts), the youth of Japan will rise against the grownups, resulting in a civil war which more or less wipes out the nation but the 2030's of COM.

Battle Royale and The Hunger Games are in the same continuity
BR is set in (novelverse and mangaverse) an AU where Japan apparently won World War Two, or Twenty Minutes Into the Future in the movie. In either case, this is several hundred years before The Hunger Games starts...but the government of Panem could have remembered the BR Program, and modeled the Hunger Games on it.

Battle Royale and A Clockwork Orange are in the same continuity
The Republic of Greater East Asia have the Program as a means to control the criminal youth as well as possible insurrection. The United Kingdom have the Ludovico Technique for that.

Battle Royale and Kill Bill are in the same continuity
Gogo Yubari could easily be the winner of a previous Battle Royale, and have either kept the weapon she was randomly assigned (the meteor hammer) or obtained another one after going into O-Ren Ishii's service.

The Selection of which class is a part of the program is not random
The class that is selected is always the one that is determined to be the one most likely to produce someone that shall grow up to be a potential threat to the regime. The BR program insures that almost all of these students are killed and that the "winner" is so traumatized by his ordeal and so marginalized by society that the odds of him ever accomplishing anything are zero.
 * Almost certainly true. When  hacked into The Program's website,   found a list of all the classes selected to participate, thus.
 * Considering how many unique freaks there are in this class -- at least half of them are completely insane or became so with just a little push -- odds are good the Powers That Be choose it because it would put on the best show.
 * Maybe in the movie but the novel and manga have the Program as something scene only by the people in charge of it and even they mostly see it as blips on a GPS and recorded conversation not much of a show.

There is a small group of schools, no more that a few dozen or so, that are offically exempt from the BR programme
These schools are reserved for the childeren of the Greater Republic of East Asia's Ruling Elite.
 * Completely wrong. It was stated several times during the manga that EVERYONE, as long as they're of appropriate age, can be selected for play, no matter if they the child of a cubicle worker or the President. One of the contestants' father held a high government post, but that didn't save him.

The current Tokyo Governor Shintaro Ishihara might draw inspiration from the Battle Royale Program
He's declared his hatred for Otaku and Gay people, but this article makes me think he would implement the program if he could.

If The Hunger Games film is successful, then the film for Battle Royale will be exported as a Hollywood Film
The only difference between The Hunger Games and Battle Royale is that in The Hunger Games the tributes aren't high school students, and they exposit more on the setting. The movie for Hunger Games might be given advertisements as if it was a Hollywood film. If successful, someone would notice Battle Royale and decide to export it as well.
 * Confirmed, sort of. The first official DVD release will be out in late March when The Hunger Games movie comes out.

The cat killed by Kaori in the manga wasn't real
Even if there were cats on the island, they should have been stray, not the cuddly variety. And stray cats avoid people. Especially if they've noticed that said people are loud and violent. Seeing how Kaori is completely insane, her hallucinating a kitten seems not unlikely.

And the cat Noriko's seen earlier was actually... a squirrel. Yes. Most definitely.

Battle Royale is set in the same universe as The Long Walk.
Both are a last man standing tournament for youths with one survivor being the victor. The only difference really, is that The Long Walk is more a battle of attrition.

The original purpose of the Program was to generate Stand Users.
If you don't have at least a piece of one of the Arrows, your only options for getting Stand Users are (a) be lucky enough to find someone born with a Stand and sympathetic to you, or (b) put the right person through enough stress that the Stand spontaneously awakens, and that's still a remote chance. So, if someone like Yoshihiro Kira (Dio and Enya clearly approved of him if he was given an Arrow by them, even if that couldn't have been until the 1980s) were in the 1940s Japanese military or government--he's about the right age--and knew about Stands (probably meaning that he himself was born with a Stand, and was in something like Unit 731 with lots of desperate guinea pigs whose own Stands might awaken otherwise unknown), he could come up with this way to both keep the populace too fearful of each other to rebel and make a slowly growing contingent of Super Soldiers via stress/terror/etc. evocation. Ideally, Stand-using survivors (anybody with a Stand is going to have a huge advantage over more normal classmates, so the odds are against an awakened Stand User losing) will end up being just broken enough to no longer be averse to killing, but not so broken/vengeful that they're more of a threat to the bakufu than the bakufu's enemies.