No, the airbag warning light in a 2007 Dodge Caliber (and most other vehicles) doesn't *need* to flash when you first start the car to indicate proper installation. It should illuminate briefly as part of the self-diagnostic process upon startup, then turn off. A constantly illuminated or flashing light indicates a problem within the airbag system. A flashing light usually suggests a more serious or critical fault.
The light's initial illumination is a check; its subsequent turning off is confirmation that the system is functioning (at least initially). If the light stays on or flashes, it's a sign that you need to have the system professionally diagnosed and potentially repaired.
