Why didn't Claire just shoot Peter?

I couldn't sleep last night (I need help...) thinking about why didn't Claire just shoot Peter; the more important question is: Peter can fly too, why didn't he fly up there himself? I think it has something to do with Nathan's need to redeem himself - he almost went along with Linderman's plan and let thousands (millions?) of people die. He swung back and forth on that before; now that he has come to power as a congressman, he probably needed to make sure he won't make the same mistake again. That's probably it.

Another question: Can Sylar be killed? He's been shot before, by Parkman, remember? He fell and then disappeared. You don't just recover from a bullet wound. There has got to be something about Sylar, that he can heal a little somehow, we should worry about that.