Can someone have a sensitivity to wheat or gluten without having an actual allergy or diagnosis of celiac? If so, what would the symptoms be?

Maybe. Someone can be intolerant to any ingested material if it triggers reproducible symptoms like abdominal cramps, loose stools, etc. With wheat there are diagnostic problems. It is a composite of many separate proteins, each of which could create an allergy. Tests for gluten/celiac are only accurate if you consume them in the weeks before testing. Re-testing with proper preparation is often needed.