No, Superman is a DC comic character and the Avengers are Marvel. Unless there is a cross over book I have not read then no.
Superman is from DC comics, movie rights owned by Warner Brothers. Avengers are all from Marvel comics, movie rights owned by Disney
Yes, he kills Doomsday while Doomsday kills him.
No- he just goes into a coma after throwing the giant rock out into space because it contained too many fragments of kryptonite which had badly infected him. It is likely he revived the way he did...
Lex Luthor- his greatest enemy of all.
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The Origins of Superman
New York Comic Con 2008
The Origins of Lex Luthor