Question: When Kara's shown putting on her Linda Lee disguise for the first time, how does she do this?
Answer: It's never explained, she could have changed at super speed. In Superman II, when confronted by the super criminals at the Fortress of Solitude, Superman did a disappearing and reappearing act. So did the criminals. It might be a similar thing.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.