I've made a few suggestions below (in brackets), but have tried not to change your text too much:
I pride myself on being calm and level-headed. You can always see me smiling. I think nothing can be solved unless you have a cool head.
(Some people call me a perfectionist). I sometimes drive my teammates mad, but I think perfectionism is a good thing. Nothing can be perfect, but at least you should try to do your best.
Another (of my positive qualities) is my cooperative attitude, as well as (my) sense of responsibility. (My) sense of duty (towards my work) pushes me to try my best in all fields, while (my) sense of duty (towards) my team drives me to care for each of my teammates and bring out the best of our ability through teamwork.