Well, to me, that particular scene is part of the amazing ability of Jo Rowling; to find not only scenes that define peoples' relationships with others--nobody can deny that that scene clearly shows that Sirius absolutely despised Snivellus--but to also (unintentionally, at times) demonstrate the true realism in the series. These people, even as children, are not at all innocent, naive or kind; they are flawed, irrational and (sometimes) murderous beings. Apart from a few truly wonderful people (e.g., Remus [who deserves sainthood for still being Sirius' friend despite the aforementioned incident] and Tonks [who, you'll note, unlike everyone else is NEVER portrayed as vengeful or willing to compromise her morals or integrity]) almost everyone in this universe is willing to do unforgivable things and make terrible choices (another example for you; Hermione mind-wiping her parents [which, while it may have saved their lives, is also a terrible abuse of her abilities] or Harry casting Unforgivables). you'll also notice that the truly good and wonderful people are quite often the ones who die, while those with flexible moralities live to bend the rules another day.There's also something to consider about Snape with regards to his time as a student; the whomping willow werewolf incident.
Snape's supporters always quickly (and rightly) point out that Sirius telling Snape how to get past the willow was a terrible thing to do. He betrayed Remus's trust and nearly killed Snape. What they never bring up is the fact that Snape willingly followed his advice. He'd seen Remus going to the willow with Madam Pomfrey, so he knew that an adult in a position of authority was involved and thus that he wasn't up to anything wrong, which dismisses any laudable motivations. At best he wanted to find out a secret of Remus's that he could use against him and there are even less innocent possibilities as well when you consider that he was willing associating with a group of violent racists at the time.
Now that I have to disagree with. Partly because I don't recall ever seeing that bit about 'impressing' Lily. If I've missed it, well...[places on duncecap] But I think he went to the Death Eaters because he felt accepted, safer there; recall what his quite pathetic excuse for a homelife was (sidebar: also consider that with his ill-fitting clothes, borderline abusive homelife, and love of not just the magical world, but the escape it offers, young Severus wasn't a far cry from young Harry; the only real difference is that of their choices) not to mention that he despised his father, and all signs seem to point to that being a mutual feeling. Plus I get the suspicion that his mother probably was spouting those anti-Muggle sentiments that Severus became endeared to. I think that being a product of that home just about permanently set him against Muggles. And think about how, after being a scraggly outsider for the first eleven years of his life, he could fit in with people, popular people, powerful people, who shared his views on Muggles. Who wouldn't have signed up for it?As for Lily giving up on Snape, given his ambitions to join said group of violent racists and the fact that it happened towards the end of their fifth year means that she had the patience of a saint with regards to him. Word of God is that he thought that if he became a powerful dark wizard, she'd be really impressed by him. For someone who was so obsessed with her, he didn't seem to understand her as well as he should have.
I think that if something little were different, like if Severus might've had a better home life, or if he knew without question Lily would be a Gryffindor, he might've turned out differently. Or, to go for one of those crackish ideas that fanfics like to offer up, if he were set up somewhere else, without the Pureblood bigotry (that dear old Eileen Prince most likely passed on to him) or the seduction of popularity (with the Death Eaters) he would've turned out to be someone that Lily could've loved. But that isn't the real world, nor is it the one that this story is set in, so I have to say that Snape was most likely doomed to being the Slytherin that he was for twenty years.
Most likely what I have to dispute is the notion that Snape had it in his head that he was going to be with Lily. Severus was many things, but he was never naive, and he was never an idiot. I don't think that he ever believed he really had a chance with Lily, and knowing that, when faced with a choice between Lily (who would most likely never be anything more than his friend) and the Death Eaters (who offered power, popularity and belonging) I think he made the logical choice...never mind the fact that it probably tore his heart straight out.
Granted, if Lily had offered her love, I think he'd have left the Death Eaters and the mindset so fast it would make Apparation look slow. But she never would, and really, never could. In terms of that, think about some older British characters, like Sidney Carton and Lucie Manette; Severus' loyalty to her and undying love for her certainly echo Carton, if nothing else. Not to mention the parallels with the impossibility that she might ever love him, and that her love is meant for his opposite, his negative. Most of all, in the confrontation with Dumbledore (during "The Prince's Tale") when presented with the possibility of having Lily, her husband and son out of the equation, he chooses her safety and happiness over his own.
Lastly, I'm tickled a bit by the "Snape supporters" bit. I suppose I might fall into their camp, though it seems that unlike some of them, I have no intention of forgetting his faults (like, as you pointed out, going into the Whomping Willow. Or uttering the word "Mudblood" to Lily). I just find the irony in this whole situation delicious, because had you told me five years ago that I'd not only be defending Severus Snape, but being a vocal and outspoken defense at that, I'd have laughed straight in your face.
I wonder if that might've been in Rowling's plan as well; let us hate the man (loudly and openly, or secretly in the dark corners of our hearts) for more than ten years, and then show us that, like Dumbledore, she "never revealed the best of him," and let us decide for ourselves if he could be forgiven, be considered good or not. Anyone else feel that way?
[Addendum: Apologies for the novella, there. My mind sometimes gets away from me when it wishes to express itself]