Now several days since I have been able to access the internet on my laptop. Finally went through the challenge of phoning Orange yesterday. Call took 50 minutes or so. Very helpful and patient as I scrambled around on the floor looking at the Livebox etc.
Diagnosis? Everything seems to be working fine, so must be the browser (IE8). Now I have to somehow find/get another copy.
Now several days later. Internet still doesn't work, so this (among other things) has forced the belated decision to finally buy a new laptop. This, at least, is a relief, though I am still to be convinced that the laptop is the source of the internet problem.