Yes, she is Princess Celestia's adopted niece.
Princess Celestia
So far there has been no mention of how Luna and Celestia came to be.
Princess Luna became jealous of Princess Celestia and tried to take full control of Equestria and bring everlasting night. Celestia was forced to banish her to prevent this.
Yes she is.
Like her sister Princess Luna, Princess Celestia is what the fans call an alicorn. Offically the princesses are all three races at once, equally an earth pony, pegasus, and a unicorn.
No. From the first episode: "...the eldest used her unicorn powers to raise the sun at dawn; the younger brought out the moon to begin the night."; this indicates that Luna, the one who controls the moon, is the younger of the two princesses.
It has not been mentioned anywhere how the princesses came to be.
Anne Heche
Her student is Twilight Sparkle who finished all her challenges about friendship, and now is an Alicorn just like princess celestia.
A princess is a kings daughter. When a king has a daughter she is a princess.
Not that we know of. Luna is her only known sibling.