Is it true that if I get a Building Permit and the work is complete and I do not call the Building Department office for a final inspection, I will not receive an Added Assessment bill?

No. The final inspection by the building department does not determine the date of assessment. The Assessor must value the property when that property is "substantially ready for its intended use", which is determined by the Assessor and may pre-date the date used for the Certificate of Occupancy or the date of a final inspection.