He doesn't want to eliminate all the dragons. Don't get me wrong, he wouldn't care if he did. His main objective is to cause as much suffering as he can, so that he can cherish the memories of his terrified opponents.
(the following answers are based on my own headcanon)
Spike's egg wasn't accidentally separated from the nest, Cimmerian accidentally didn't damage it when he smashed the clutch. Neither he nor Spike's parents noticed this, however, since this was only visible from a certain angle.
Jade and Spike are friends.
Flicker is Spike's brother, but he doesn't hate ponies. He just thinks they are weird.
Spike's mother's name is unknown. Her mate calles her his 'amethyst', but that is just a pet name.
Spike's father is called Thorn.
Why would he want revenge? He successfully destroyed Spike's family.
You have someone else's work confused with mine. I forgot whose.
Tsk, Tsk, Tsk Cimmerian, you should know better than to leave that one egg left (if you did leave it on purpose). It's always 'the last egg', 'the last child', or 'the last survivor' that ends up killing the great evil force.
Except not in this case. Well, not entirely. It takes Celestia, Luna, Discord and the elements to take him out. Spike helps on the sidelines, but it is ultimately Discord and his reality changing abilities that gives the good guys the advantage.