Assume a Contained Concepts exists. How to add the existing Concept as a Contained Concept to its parent Concept? Should a user have to obtain the count of the Contained Concepts and increase the index?

No, you do not need to obtain the count and increase the index. To append a Contained Concept to its Parent Concept, use the following catalog function:

Instance.PropertyArray.appendContainedConcept();e.g.
 

Concepts.Container co = Concepts.Container.Container("C_001","Container_001","C_001");
Concepts.Shipment sh = Concepts.Shipment.Shipment("S_001","Shipment_001",null);

long addDate = DateTime.getTimeInMillis(DateTime.now());
Instance.PropertyArray.appendContainedConcept(sh.Container,co, addDate);