Sirius was trying to teach a spying, sneak a lesson. I am convinced he is the one who told James what he'd done, which then sent James after him to save his life.
Do you think that sneaking around and trying to get students in trouble for activities that they shouldn't have been doing deserves death or lycanthropy?
I don't think he actually meant to kill Severus, because he cared too much for his friend to allow him to live with the guilt of biting another human
I'm not so sure about that. In fact, Sirius not caring about Remus enough would explain quite a lot that happens later on. What else would make Sirius and James suspicious of Remus unless something came between them? And what could have been bigger than the werewolf prank? Remus suspected Sirius to be the traitor-- if he, being true to himself, didn't protest too much about the prank aloud, because he was afraid they would leave him, but harbored doubt, then it would explain why he thought Sirius betrayed them.