SUB-ZERO always have revenge & vengance for Scorpion,he felt that Scorpion was responsable for murdering his brother in the Mortal Kombat Tournament,so he trained the Lin Kuei Ninja Clan to become ninja assassins & the powers of ice & cold.That's why that Sub-Zero have earned the title of GRANDMASTER of the Lin Kuei Ninja Clan.
No. None of his biographies in the original timeline said anything about showing interest in knowing who killed his brother. The second game, his story has him where he's focused on trying to assassinate Shang Tsung, than worrying about Scorpion. 2011 only has it where it focused on him worrying about his brother's disappearance/death.
3 - Betrayed the Lin Kuei and not only has to worry about Shao Kahn, but the automated cybers that are chasing after him.
4 - Worries more about Shinnok and learns the secrets that could stop Shinnok that he learned from his brother
Deadly Alliance - He obtain the Dragon Medallion from Sektor because the future of the Lin Kuei was hanging on balance. He trained his students and see which one could help him aid him against the Deadly Alliance.
Deception - Discovered more of his heritage after laying Frost to rest in an ice cave after her betrayal.
I certainly must say, he's got nothing to do with Scorpion after the second game according to his story. But yeah, they'll always be rivals because the two most iconic characters would always be pinned in a fight for show.