Can early symptoms of appendicitis be present for a week?

Not really. Appendicitis generally evolves over 12-24 hours. Either the diagnosis is made and the appendix is removed, or the appendix ruptures and other treatment ensues; the appendix may or may not be removed at a later date. Symptoms that persist for a week are unlikely to be related to appendicitis, but should still be evaluated. See your doctor.
May be. Appendicitis usually develops quickly but inflammation associated with or precipitated by viral infection of lymphoid tissue or inflammatory disease of the intestine may be present for a few days prior to the appendicitis becoming manifest.