<<< automatic config on the troubled laptop reads an ip address 10.0.0.3 which is invalid for access the internet through the router. >>>
The 10.0.0.x subnet is perfectly valid for Local Area Networks and there is no reason why clients being allocated an addrees in this range by the router should be unable to access the internet. If your router operates in this range, don't try to use manually entered 192.168.x.x addresses. What is the IP address of your other computer that is getting online?