OK so there's mysteries around Eggman Nega, why he appears to be from another dimension in Rush, is portrayed as Eggman's descendant in Rivals, and why he'd want Eggman or Earth removed when that could threaten his own existence if he is a descendant.
Well, the latest news is they confirmed that Eggman Nega is Eggman's descendant from the future, so here's what I think happened.
First, I'll address the question of how Eggman Nega can have a base set up in the future in the Rush series at least if the future is ravaged by Iblis. It could be that Eggman's son or daughter (whichever it is) resides on another planet, or some kind of space colony like the ARK, and was therefore unaffected by Iblis. This would also explain why Eggman Nega would want to trap Earth in a card. Presumably the Robotnik line continues on this planet or colony until it reaches Eggman Nega. Whether they're all as insane as Eggman is unclear, but Eggman Nega certainly is.
Now, as he says in Rivals, he wasn't respected as a scientist because of the failures of his ancestor, Eggman. So he decides to try and change that name by redeeming Eggman's failures and helping him rise to power. Having somehow learned of the Sol Dimension, he uses his time travel device and interdimensional transporter to go to the present (for the sake of an alliance with Eggman), and enters the Sol Dimension to take its Sol Emeralds, with a plan for creating Eggmanland alongside Eggman. We'll say he goes after the Sol Emeralds first, but he runs into Blaze, and she fights him to defend the Sol Emeralds.
Eventually, Eggman Nega decides for a different course of action. He returns to the Chaos Dimension and forms his alliance with Eggman. He guides him to the Sol Dimension, and distracts Blaze while Eggman steals the Sol Emeralds. Then they return to the Chaos Dimension, and, as Eggman Nega planned, the Sol Emeralds being taken to the Chaos Dimension causes the worlds to go on a collision course, with plans to destroy both worlds and build Eggmanland in its place. How the survival of Eggman's son/daughter was planned is unclear, but Eggman and Nega did plan to survive the destruction themselves, so they probably took measures. Unfortunately for the Eggmans, Blaze is also transported to the Sol Dimension, and she and Sonic defeat them and save both worlds.
Of course, they both survive, and Eggman Nega comes up with another plan, to steal the Jeweled Scepter and use it to conquer the world. They go to the Sol Dimension and create Captain Whisker to steal the Scepter for them, and after Sonic and Blaze kill Whisker, the Eggmans reveal themselves, during which Eggman Nega tries to blow up the planet with him and Eggman still on it, which is where Eggman falls out with him. They barely escape destruction when Sonic and Blaze blow up their mech, and from here, Eggman and Eggman Nega end their alliance and go their separate ways.
Eggman Nega decides from here that Eggman is too pathetic for his failures to ever be redeemed. He makes one final attempt at defeating Blaze, as revenge for her foiling his plans, but she defeats him. So, he gives up on ever beating her and makes his escape to the future. Blaze follows him. However, as they appear on Earth of the Chaos Dimension, they are immediately greeted by Iblis, who kills Eggman Nega. He tries to kill Blaze, too, but Silver appears, and he and Blaze fight against Iblis. This is where their alliance comes from in Sonic 06.
Of course, when Elise kills Solaris of the past, as a flame, Iblis is erased, so when Eggman Nega and Blaze arrive on Earth in the future, there is no Iblis and Earth is thriving. Silver forms an alliance with Blaze, they fight Eggman Nega and seemingly kill him (much like many of Sonic's battles with Eggman). Or, Blaze doesn't meet Silver but Eggman Nega either escapes or Blaze takes him out in a similar style to Sonic the Hedgehog 1 (as he wouldn't have time to set up any new mechs). The battle won, Blaze returns to the present day Sol Dimension.
But Eggman Nega is still alive, and finds a new archnemesis in Silver, who protects the camera that traps people in cards. Eventually Nega steals the camera, travels back in time, and traps Eggman in a card. A.k.a. Rivals. Nega tries to trap Earth in a card, either tricking or programming Metal Sonic to work for him, but is defeated by Sonic, Knuckles, Shadow, and Silver, and is trapped in a card himself. Silver returns to the future with him.
However... Eggman Nega has a Metal Sonic of his own in the future, Metal Sonic 3.0. He releases Eggman Nega from his card, and Nega decides then and there to bring hell on Earth by releasing the Ifrit. Thus begin the events of Sonic Rivals 2. In the end, the Ifrit is destroyed by, we'll say, Shadow and Metal Sonic, guessing they're the last ones to fight him seeing as how they weren't able to get through the portal, and presumably kill Metal Sonic 3.0 as well, because when Eggman Nega's legs are caught under rubble, he turns to his enemies, Silver and Espio, for help first. Eggman Nega is left trapped forever in the Ifrit's dimension, presumably to die of dehydration in that inferno.