Red Hood is a character who has been variously depicted in comics. Sometimes it's Jason Todd, who was once Robin and came back with a more violent approach.
Red Hood doesn't kill Hawk in the comics. The relationship between them is complex and involves various interactions, but killing each other isn't part of it. The comic writers chose a different path for their characters' development.
It depends on the specific comic arc. Sometimes Red Hood shows signs of turning good, but it's not a straightforward and permanent change. His character remains complex and his actions can vary depending on the circumstances.
His motivation could be related to a difference in ideology. Red Hood has a more brutal approach to crime - fighting compared to Nightwing. He might think that Nightwing's methods are too soft and by kidnapping him, he wants to force Nightwing to see things his way.
It could be someone manipulating Red Hood. For instance, a powerful crime lord who has some dirt on Red Hood and forces him to kidnap Nightwing. So, in a sense, this crime lord would be the real villain.