Is ultrasound necessary in determining whether I have appendicitis?

No. The diagnosis of appendicitis is made by taking a careful history of one's symptoms, doing a careful physical exam, and performing basic laboratory work. If the diagnosis is unclear, a ct of the abdomen, or, less frequently, an ultrasound may aid in making the correct diagnosis.
No. History and physical exam may be all that is needed. Commonly in the us a ct scan may be done ultrsound can be helpful in looking for other conditions : gallstones, pelvic cysts etc.
Old school. An ultrasound is absolutely not required to diagnose appendicitis. An "old school" but effective approach to make the diagnosis is to listen to the patient's history, perform a thorough physical examination, and check some basic lab work. An ultrasound can help support the diagnosis, as can a ct scan. But neither is necessary.