In addition to what I've already found out, I have now confirmed on Neo Green Hill 2 that choice of character doesn't influence randomization. I thought that the randomization might be the same, so since you hit a trigger before the ? box on Egg Rocket I thought that was influencing it.
You'll always get 5 Rings from the first random box, no matter what, if you just spindash through it or get it without hitting what are unconfirmed triggers and red springs. From what else I've seen, a lot could trigger new randomization, the prime example being the red spring. But I did 2 entire runs of NGH2 without using red springs and the random at the end changed anyway, so i assume that either hitting checkpoints and/or the poles that let you swing higher change the randomization as well.
Also you have to give the randomization time, or else it will not happen. For example if you hit the red spring before the ? in Egg Rocket and instantly go to the random box within what seems to be like 10 seconds, the randomization will not have triggered yet. It is unconfirmed if you have to wait exactly 10 seconds (since it's hard to judge with the Egg Rocket timer shenanigans and what not) or within 10 second periods (I might be using a wrong word combination here so i will give an example - I do not know whether its 10 seconds from hitting the trigger or if it's changing at 0:10, 0:20 etc., assuming you hit the trigger before then)
It might also be 5 seconds, but 10 seconds seems to be more likely.
EDIT: The assumption above is probably wrong, as the randoms seem to change as soon as you hit a trigger and the Random Box is out of sight. However there seems to be a regularness to it that is hard to describe with words. If I get to it i might record it on emulator, because I don't have a way to record from my fat DS. Results are the same on both, so I expect it has to do with the games coding, not its version.
EDIT2: It seems that red springs only change the ring box if you go far enough, just hitting it turning around and hitting the ring box seems to only affect it sometimes. I really wish I could figure out how this entire thing works.
EDIT3: It appears that time is playing a roll in what random you get, but only if you hit some sort of trigger first. If you stand next to a random for 4 minutes and then get it, it will be the same result as if you got it directly. However if you do the same path that you know has triggers at 2 different points in time, you will get different results. Now the problem here is that I don't know what falls under patterns and what does not, because it seems that if you do 100% the exact same things in a couple runs, you will get the same result in terms of random boxes. That, however, is extremely hard for me to replicate.