I am going to take a bit of a different approach to why Rick (Humphery Bogart) is cool. At the beginning of the movie Rick's Cafe America is the place to be. It is the glimmer of hope that so many cling to, and to many it may be as close to America as they come. So what do you have to do to get in? You have to know or appeal to Rick. So right off the bat we are trained to think that Rick is cool. Then some of the first images that we see of Rick are of a man sitting alone smoking. We find out that he does not drink with customers and he does not care where you are from or how powerful you used to be. Again this adds to the idea that Rick is cool, He is the guy you want to know.
Rick's hard shell and rough physique is transparent because all that it truly says about Rick is that he was hurt once or that he once stuck his neck out on the line and he will never do it again. From that moment you that, you want to empathize with him and see him develope into a softer character because we want that sort of hope for all people. The audience is now accepting of the idea that Rick is cool but he becomes more cool in the viewer's eye because they can relate to him. And why would you not want to be able to relate to Rick? Because he is a guy you what to be like .
When Ilsa comes into the picture it makes sense. He was a man hurt by love. You begin to say to your self that you know what that rejection is like. As a viewer you want to believe that something will open his heart back up again. That there is some way to ease the pain, that by watching you will be able to ease your own pain.
During the film when ever Rick is faced with this, the possibility of unrelenting happiness, he turns it down because he is willing to sacrifice his love so that another might enjoy it. It is a very virtuous notion (to me it is a cop out). Each person in the theater is now questioning whether they would do the same for some one else. They hope they would because everyone wants to be that cool. They want to be so cool that they would give up what they cherished most so some one else could have it. So that they could transcend the values of a rationally interested society.
My point is that we think that Rick is cool because we as individuals hope that we could act with the same values and courage that Rick did. We all seek the idea that there is always something more than what we want most. That at the heart of it all there is a value that will transcend personal wants.