Green Biker Dude's role is to show how far X has come. In the previous game, X was the rookie who charged in headfirst, got defeated by Vile, and was saved by Zero. Juxtaposed against GBD, who gets himself killed like a rookie, it shows that X has now become a seasoned Hunter, leading the charge in Zero's place. GBD even resembles X, as if we're watching the old, inexperienced X die and the new, experienced X fight on.
The piece of Zero that each X-Hunter keeps is directly related to their core attribute. Serges holds Zero's head (representing intelligence), Agile holds Zero's legs (representing speed), and Violen holds Zero's chest and arms (representing strength).
Magna Centipede is the most story-important of the eight bosses. First, his backstory is that he is a brainwashed Maverick Hunter, which foreshadows the X-Hunters' plans for Zero. Second, his Central Computer Stage is responsible for transmitting a virus around the world - this was the introduction of the Maverick Virus. Finally, his stage is actually the location of the final battle with Sigma.
When defeated, the X-Hunters hand over Zero's parts, despite Zero's importance to their plan. If one believes that Serges is some form of Dr. Wily however, it's clear that he doesn't want Zero to become Sigma's puppet. This also clears up his team-up with Sigma in X5; after beating Dark Necrobat, Alia comments that Sigma used so much energy spreading his virus that he may die. The real plan was to have Zero "awaken" and then kill the weakened Sigma, before dealing with X. The only reason it failed is that Lifesaver defied orders and warned X about Zero's power boost.
Serges and Agile are completely different in their rematches, but Violen is virtually the same in his. While the real reason is because Violen's second form was cut due to cartridge space limitations, this actually works for his character: it's said that Violen isn't very smart note He has the lowest speed stat of the three X-Hunters by a wide margin (reflected in his rate of speech being the slowest of the three), he doesn't take part in any of planning, indicating that Serges and Agile are the brains of the group, and when he does speak up he only makes simple and obvious statements., so a dumb brute like him wouldn't think to switch up his primary form of attack for a rematch and would stick to what he knows.
If Fake Zero is supposed to be Zero's clone, why is it so weak and colored black? Because it only appears if X gathered all of Zero's parts. Without any of Zero's parts, the X-Hunters will have to make do with what they vaguely remember about his design and rush its production so quick that it doesn't even get a proper paint job. This also ties to its later appearances in later games as an 'armor' for Zero where it reduces his defense (namely X8). Black Zero really can't take a hit.
Sigma was weak to Electric Spark in X1. This time around, he uses attacks very reminiscent of both the normal and charged versions of Electric Spark. Seems like he took that weakness personally.
Wonder why those spotlights exist in Magna Centipede's stage? They are indeed used to keep X out but, their primary function is to keep Magna Centipede in, ensuring that even if somehow Magna Centipede managed to break free from Sigma's influence, he would still be stuck working for Sigma against his will.
