I felt that this idea (Edward being able to suck venom out of Bella's blood) to be something of a paradox.  In my opinion, the author means it to be perplexing, causing the reader to ponder how a love can be so strong to resist such primal urges.  This thoughtfulness on our part causes us to dwell on the strength of the love between Bella and Edward even further:  one of the goals of the novel.

Edward has to prove to himself that he can maintain control with Bella no matter what. If he is unable to save her life now by controlling his instincts, then the best thing for both of them is to never see each other again. Besides, Carlisle is busy trying to patch her up.

I think the answer is found in the author's religion.  Mormons beleive very strongly in "agency"--the concept that the world is full of evil and a person has the right and ability to choose evil at any point.  It is personal choice and agency that makes a person decide to do good in the face of all that temptation. Here, Bella's blood is the ultimate symbol of the temptations of the world.  Edward wants it... he craves it and one moment of weakness will have him draining her.  However, if he can choose to do the right thing, he will be even more rewarded at the end.

Edward has deep feelings for Bella, his ability to resist her blood stems from the fact that, first of all, he doesn't drink human blood and because he loves her and would never consider drinking her blood.  He reveres her, she is not a food source.

Carlisle did not suck the venom out because as a doctor, he was working on stopping the bleeding from her head wound.  

Carlisle was trying to stabilize her as a doctor, so he would not be able to suck out the venom that is why Edward had to do it becore it was too late.