1. Korra has never entered the avatar state. Now, I’m hoping that the way things are starting to set up about her connecting with her past lives that we’ll see some avatar state soon.
2. What happened to the rest of the world? Did everyone all move to republic city? What’s happening in the fire…
1) It has been stated that Korra excels in the physical aspects of bending but not the spiritual. She even has problems meditating. She also hasn’t mastered or even shown ability in airbending at all, therefore making Korra unable to enter the Avatar State. In the coming episode, she finally contacts Aang, hopefully bringing her one step closer to reaching the Avatar State/entering the Spirit world etc.
2) The Legend of Korra was initially supposed to a 12-episode mini series, thus its fast-paced and concise nature. Bryke has expressed that the setting of Korra as a miniseries would only be confined to Republic City. After the production of the first season was complete, Nickelodeon requested that they produce a second season. I’m assuming that the Krew travel to some extent in the second season since there were some released storyboards showing Korra and Mako in the South Pole.
3) Korra discovered that she was the Avatar at the age of 4. The Order of the White Lotus then took Korra from her parents to a recluse compound for her to commence her training as the Avatar. From the ages 4-17, Korra is visited by the various bending masters who train her, instead of her travelling the world like Aang did. (Her confinement was due to a request from Aang for her ‘protection’). Her most recently mastered element is Fire, as clarified in the first few minutes of Episode 1, and Tenzin was initially supposed to live at the compound to train her in Airbending, but because of his commitments in Republic City, Korra has to visit him instead to receive her training.
It’s all already been explained the the show, I guess you had to pay a little more attention haha. Hope this clears things up :)