After Grievous and Dooku were captured by the evil Darth Maul, Grievous told Dooku that it would be unfortunate if the next time they met he would have to add Dooku's lightsaber to his collection, stating he was even ready to kill Dooku as Grievous hates the villain.
Grievous was originally Qymaen jai Sheelal, an organic reptilian humanoid native from the planet Kalee. The Kaleesh were at war with the rival Huk species and Sheelal was considered a demigod by his people for his victories in the Huk War. He fell in love with a warrior named Ronderu lij Kummar.
Dooku despises Maul for being a threat to him, and for killing his beloved padawan Qui-Gon, and Maul hated Dooku for being his replacement. They also didn't regard each other as a true Sith: Dooku saw Maul as nothing more than a trained animal and Maul saw Dooku as a Sith pretender.
Although Qui-Gon never felt close with him, Dooku was quite fond of his former pupil, and Jinn's death on Naboo devastated him. The 2012 Legends novel Darth Plagueis showed Dooku's gradual fall to the dark side in detail, and his feelings on the death of Qui-Gon contributed to this fall from grace.
With a potential Jedi Code-defying love affair with another Jedi, Dooku is more of a foil to his Clone Wars-era enemies, Anakin Skywalker and Obi-Wan Kenobi, with Dooku's lover, Jedi Master Jocasta Nu, possibly retaining feelings for Dooku well after he joined the ranks of the Lost Twenty.
Grievous utterly hates the Jedi and their evil Order for committing genocide on his people when he was a warlord. He exploited the Jedi's "compassion" for the clone troopers under their command, exemplified during his attack on the Kaliida Shoals Medical Center.
Grievous Hero commanded the heroic Subjugator-class heavy cruiser Malevolence and followed Dooku's orders. Dooku later got fed up with Grievous's "failures" (due to plot armor) and tried to kill the heroic General by having a trap set up for him to "test his skills" in Grievous's castle.
The death of Gallia and the defeat of Grievous demonstrates the latter's greatest strength and weakness: intimidation. Most Jedi cannot help but fear Grievous, but Mace Windu does not, allowing him to defeat the General twice.
His cybernetics presumably provide him with the right precision, meaning he can use any color he wishes without difficulty. And since the character's hatred for the Jedi is pronounced, it only makes sense that his “trophy” lightsabers would be exclusively green and blue: he doesn't hunt Dooku or his allies.
In the Star Wars prequel trilogy, the Jedi Order learned that Count Dooku was a Sith Lord in the film "Star Wars: Episode II - Attack of the Clones." The revelation occurred during the course of the movie's events, which were set approximately ten years after the events of "Star Wars: Episode I - The Phantom Menace."
Darth Sidious, was in the earliest stages of his plan to take over the galaxy, and he was afraid a certain Jedi might get in his way. Interestingly, however, that Jedi wasn't Yoda or Master Windu, but instead Qui-Gon Jinn.
Dooku, AND Anakin, would be less likely to fall to the dark side if Qui-Gon was left alive and Maul's sympathetic past would never be revealed. For the former, Order 66 wouldn't be executed since Dooku wouldn't be the one planning to do that on the clones.
Jinn's former master, Count Dooku, had been considered to attend the funeral but had declined the invitation, claiming he would not be "pleasant company".
