I heard that appendicitis can resolve itself with mere antibiotics. What is the real danger, then?

Serious infections. Once appendicitis begins, it will inevitably rupture, with or without antibiotics, in a matter of days. Once the appendix ruptures, it is unpredictable whether the hole will seal on it's own, lead to an abscess, or cause a life-threatening infection. Under most circumstances, the risk of surgery is much, much lower than the risk of infection.
No. Once appendicitis starts it progresses and worsens and eventually ruptures. In some instances an appendix that has been ruptured for awhile and has an abscess is drained by cat scan and treated with antibiotics and then removed 6-8 weeks later when all the inflammation has resolved. This is called an interval appendectomy and can still be done laparoscopically.