I had an appendectomy 6 days ago and I am still feeling sick to my stomach/nauseous. Not all the time, but once in a while. Is this normal?

Postop appy. Some postop nausea is common post appendectomy. If you can eat without vomiting, are fever free, and you are defecating, I would say you are likely fine. I imagine you are taking some pain drugs which contain narcotics and that can also cause nausea. Discuss the situation with your surgeon. Try some ginger tea and stop the pain meds.
No. You should be feeling better day by day. Check in with your surgeon for a reassessment. All the best.