Kate tells Sun in "Hearts and Minds" that she was actually heading to Bali, perhaps revealing that being in Australia was only an intermediate step. She had very little incentive to lie given that she did not believe Sun spoke English at the time.
Her true feelings lie with Sawyer, because when Sawyer told her to leave when they were in the cages she could not, and climbed back into her cage. However, when Jack tells her to leave she goes, because Sawyer is coming with her.
That's quite simplistic. She didn't run away when out of the cage because she obviously didn't want to leave any of them behind, and also cause she probably knew that she couldn't make it back on her own, they didn't even know where on the island they were (as it is, she didn't even know that they were on another Island).
Cassidy sent Kate money and informed her that Diane was dying (after she turned Sawyer in, her money was returned to her).
In "Par Avion", Mikhail states that Kate is not on Jacob's list because she is "flawed". This may be a reason why Kate is not pregnant; she is unable to conceive, tying in with The Others "breeding" program.
She may have used the ticket Kevin got for their honeymoon, traded it in for a ticket to Australia. Then after she was turned in by Kevin they traced her fake name to the ticket and to Australia and that's how they knew she was down there and put up posters and a reward.
Kate Used A Fake Passport
During Kate's trial for murdering Wayne, the judge offered her 10 years of probation and confinement to the state of California, meaning she can't leave California for 10 years. She agreed, yet she went on a plane to Guam. I think she either used a fake passport to get on the plane, or she made some kind of deal with the judge, that we just didn't see.
She is not suppossed to leave the state but she's not being monitored 24/7 nor do the airlines get that kind of info so using a fake name or passport would not be neccesary.
Why Kate won't allow her mother to see her son
Because Kate is unable to relate to other people. All she ever wanted from her mother is her to approve of Kate's murder of her father. She has no other emotional attachment to her mother.
Because Kate thinks she will know (by intuition or other means) that this is not Kate's biological child.
Because Kate feels that her mother is blackmailing her by only agreeing to not testify against her in exchange for getting to see Aaron. Kate is still mad at her mother, and does not want to give into her, nor give her the satisfaction of seeing her grandson. Kate is acting out of spite.
Kate's mother was not a positive influence in Kate's life. She stayed with an abusive husband who leered at Kate, then essentially chose that husband over Kate by turning her in. Kate does not want Aaron exposed to someone she feels has betrayed her repeatedly.
Kate is innocent in the flash-sideways
When talking with the mechanic, she says, "I'm wanted for murder.", and also asks "Would you believe me if I told you I was innocent?". Kate seemed very sincere when she asked Claire if she would believe that she was innocent. It is unlikely she would have asked this question of a complete stranger with nothing to gain.
Between Season 5 and Season 6 the America's Most Wanted commercial featuring Kate said the gas leak and explosion did indeed result in murder, but not of her step-father - of his assistant. Now, since this is a sideways universe, her mother could be blamed instead of her, and Kate is just taking the wrap for it.