I think the reason Carlisle didn't suck the venom out of Bella's hand was because Edward was too territorial with Bella. I think maybe Carlisle wanted to prove to Edward he was strong enough to save Bella himself? And if none of those answers work, then maybe he didn't do it just cause SM thought it was more impactful for Edward to do it himself?
As far as Edward's eyes not turning red, I think it was discussed before, or maybe somewhere else, but I believe it was determined that he didn't take in enough of her blood to turn his eyes. He's been a vegetarian vampire for so long, it would take more than just that little bit to make his eyes turn red again. And it could be possible that his eyes did turn slightly more red right afterward, but by the time anyone saw him (like Bella or her parents) his eyes would have been back to normal already (assuming his system processed the blood quickly since it was such a small amount). Anyways, that was the idea