"He has a style."
1. What does it mean
It probably refers to his musical style
"I felt all flushed with fever."
2. Is it OK to say, "I felt so embarrassed. I felt all flushed with fever?"
Not in daily language...there's a lot of poetic licence used in song lyrics. If you heard someone say they were flushed with fever, you would assume they were ill and had a high temperature
"But he just kept right on..."
3. Does 'kept right on' mean 'kept on' or continued? Does right have a meaning here?
Yes that's right. "right" kind of underlines it
"He sang as if he knew me. He looked right through me as if I was not there."
4. Do 'as if ' and 'as though' mean the same thing? Is it OK to rephrase the quote above and with no meaning changes, "He sang as though he knew me. He looked right through me as though I was not there."
He sang as if he knew me. (In fact, he didn't know me.)
He looked through me as if I was not there. (In fact, I was there.)
First line is okay, second line means that he did not notice her
5. What does 'look through someone' mean? Is it a common usage? Is it equivalent to 'see through someone'?
"To look right through someone" means to ignore someones presence
"To see through someone" means to means to see the hidden truth behind what they are saying.
"In all my dark despair"
6. Do native speakers say this often?
It's rather dramatic, but correct. She is trying to get you to understand that she's really depressed/in a hopeless situation