화(가) 나다 (get angry), 화(를) 내다(get angry at someone), 화를 참다(화 나는 걸 참다 - suppress one's anger) are the only common expressions. I've never heard of 화를 쓰다.
쓰다 (work up) is used in certain emotion-related expressions, but not for 화.
- 애쓰다 / 애를 쓰다 - try hard to get something done.
- 떼쓰다 / 떼를 쓰다 - act up childishly to have one's way.
- 용쓰다 / 용을 쓰다 - muster up all one's strength to do something.
- 인상(을) 쓰다 - make a face in displeasure or to show annoyance.