Green stool with mucus would indicate that you have a stomach infection. You need to go and see your doc and get a stool sample tested. Have you been taking antibiotics recently? If so, green stools can indicate a C.Diff infection due to the antibiotics.
Anything with itis on the end of it means inflammation, so colitis means inflammation of the colon and inflammation by definition will cause diarrhea rather than constipation. I have never heard of anyone with ulcerative colitis who does not suffer from severe episodes of diarrhea during a flare up. Due to the inflammation this diarrhea normally has blood in it. Ulcerative colitis is thought to be an auto-immune disease, where the body starts to attack itself and this causes inflammation and ulcers in the colon. Most people have their condition controlled with steroid medication to control the inflammation, antibiotics for the infection and sometimes immunosuppressive therapy to reduce the immune response of the body. In severe cases though, a colectomy is the only option and a person will have to live with an ileostomy for the rest of their lives.
From what you have said, it does not sound as if you have ulcerative colitis. I would check out the infection possibility first and try and sort your constipation out. If you did have UC you would most definitely be having diarrhea. Little round and hard stool indicates constipation big time that you need to deal with.