no
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No you don't. I caught it after I beat the Elite Four. You can catch it before or after. It doesn't matter.
You don't necessarily have to beat the elite four to catch any event pokemon, especially since the only way to get Arceus right now is to hack it.
Yes. You also cannot catch Superjesuslatios or latias without the elite 4.
After you beat the elite four and the galactic biulding
If you beat the elite four before you catch raquaza you can use the cloning glitch to get as many master balls as you want (the reason for waiting until after you beat the elite four is because the cloning glitch takes place in the battle frontier