Since when and why does Myrcella Baratheon from Game of Thrones live in Dorne rather than with her parents in King's Landing?
Tyrion sent her to Dorne as betrothed to their prince. She was sent as ward/hostage to get Dorne alliance.
Refer Season 2 in GoT wikia:
In order to strengthen House Lannister's political position during the War of the Five Kings, Hand of the King Tyrion Lannister makes plans to forge a marriage-alliance with House Martell of Dorne by having Princess Myrcella wed to Trystane Martell, the youngest child of the current ruler of Dorne. In return House Martell would hopefully supply soldiers to the Lannisters' side, or at least ensure they wouldn't openly side with their enemies.
The actual marriage ceremony will wait until Myrcella is old enough, but part of the arrangement involves sending her to Dorne to live in Trystane's household. Queen Cersei is furious with Tyrion when she hears (from Grand Maester Pycelle) what he is planning, declaring that she won't let her only daughter be shipped off to Dorne as she was shipped off to Robert Baratheon. Tyrion counters that as a princess, many would say that Myrcella was born to one day forge a marriage alliance, and she will be safer in Dorne than in the capital city during a major civil war. Against Cersei's continued protests, Tyrion insists that the plan has already been finalized, leaving Cersei fuming and in tears for her daughter.
It was in episode S02E03 that Tyrion has the idea to send Myrcella to one of the other houses and later marry her to one of their sons. This is on the one hand to get her to temporary safety out of King's Landing and on the other hand to make a stable alliance with the respective house in the future. And this idea is then also put in motion during the rest of the 2nd season and Myrcella is sent to Dorne.
He actually tells three different plans to Varys, Petyr and Grand Maester Pycelle respectively, in which he changes the house with which to marry Myrcella, be it Arryn, Greyjoy or Dorne. It is however the story about Dorne that gets told to Cersei by Grandmaester Pycelle and which is then put in action. While it is not entirely clear if that was his plan all along or just chosen because it was the one revealed to Cersei, it is clear that Tyrion always planned to marry Myrcella to one of the other houses and have her live with them for the above mentioned political reasons.