Why she was not sent back in time like the other Oceanic 6'ers?
Sun didn't go back in time because she was not a candidate. There is strong evidence that Jacob and his brother are based of the story of Jacob and Esau from the Old Testament. In a passage from the Old Testament, Jacob wants to pass his convenant through his grandchildren, and touches Manasseh and Ephraim. However, he touches Ephraim with his right hand and Manasseh with his left, proclaiming Ephraim will be a greater person. Similarly, Jacob touches Jin with his right hand and Sun with his left, only making Jin the candidate.
Sun wasn't sent back in time because she "betrayed" the Island by working for Charles Widmore just like Ben wasn't sent back in time because he moved the Island when Locke was suppose to and then even worse he came back
Sun wasn't sent back in time because she was carrying a tracking device for Widmore. That is how he planned (successfully) to return to the Island. The tracking device worked as a sort of tether in time.
Sun didn't travel back in time because she was originally not supposed to be on Oceanic 815, and thus not supposed to be on the island in the first place. If you remember, she was originally going to leave Jin at the airport, but it was through her free will that she stayed with him. The island made sure she got pregnant so as to force her off the island so she would not travel back to 1977. She was supposed to leave, but Jack, Kate, Hurley and Sayid were not.
Wasn't she supposed to leave in LAX?
She did not successfully replicate the conditions under which she originally went to the island.
She was not supposed to have returned to the island. She was a mother and should have stayed with the child. She returned and it led to her death (Jin had the free will to escape, unlike her).
She was never a candidate. Only one of the Kwon's was a candidate and that was Jin. That was the reason she didnt travel in time. She was just like any of the other passengers on Ajira.
Also, another reason that Sun isn't the Kwon candidate is that Rose was listed as Henderson on the candidate wall, Henderson is her maiden name, so who's to say Sun isn't her maiden name, Paik.
Then why did Ben, also believed to be a candidate, stick with Sun? (Unless "Linus" refers to Roger, and considering Ben is an actual leader, while Roger was a drunken, abusive father and a janitor for all attempts and purposes, this is highly improbable.)
I think you mean intents and purposes.
Sun was supposed to leave the island and raise Ji Yeon alone
Jin and Sun were supposed to get to the island, so that Sun could become pregnant. After Sun became pregnant, she and Jin were destined to be apart. They were separated when Jin left on the raft, then again when she was almost taken by the others on the Elizabeth, then again when the Kahana blew up while the Oceanic Six escaped. This could explain why Sun didn't travel back in time with Jack, Kate, Hurley, and Sayid. She wasn't supposed to return in the first place. This implies that the candidate "Kwon" was actually Jin, and that at least part of Jin's purpose was to be on the island alone in Dharma time to identify the pockets of electromagnetism that he shared with Widmore. Sun and Jin made the choice to fight this destiny and die together rather than live without each other.