The PC's IP address is 10.0.0.138, and the local gateway is 10.0.0.1, which looks normal for a NAT setup. The PC reports
its connection is OK, and if you Ping the gateway, I bet it will work:
ping 10.0.0.138
Then, if the IP connection is fine, but you still have problems connecting to a name, then it's a Name Resolution problem. I see that the report seems to have the correct value in there:
DNS-servers ............................ : 10.0.0.138
And when you try to ping BBC, you get
Answer from 10.0.0.138: Goalhost can not be reached.
That is, your gateway is reporting that
it cannot ping the BBC.
Since your gateway and PC are communicating, I think your gateway or your ISP are having problems. I don't know what to make of the fact that other machines work - cached DNS records? Have you tried restarting the connection on the gateway, or the gateway itself?
re fixed IP addresses, you should not need to resort to such measures on a home network, unless you enjoy that kind of thing! DHCP is mature enough to cope, these days.