the click noisy is probaly a relay turning off this should make this click when it turns on and off, this is perfectly normal most pc's/laptops have them.
About the pc turning off im right in saying this laptop has an in built battery so that wouldnt be a problem. But if the powerpack is faulty and it hasnt be charging the battery it may turn off straight away when the powerpack fails. Try a different powerpack!
About the machine shuting down on its own, this maybe a cooling problem, how hot is the machine when it does a shut down?
And make sure the power saving setting in windows are set properly.