If an object is not required, do I still need to include data for the parameters listed as "required" within the object?

No. Parameters in an object that are listed as "required" are only necessary if you include the object in your call.

Until what stage can the details of an order be changed?

You can send the Update Order request while the order has the New status. Once the status has changed, it is not possible to update order details.

If an order or fulfillment has multiple shipments, which delivery status does Delivery Hub display?

Delivery Hub always displays the status that is earliest in the delivery flow for a list of shipments. For example, if a fulfillment has one shipment that is ready for carrier collection and another that is out for delivery, the fulfillment or order is assigned the ReadyToShip status.