I am not giving up, although discouragement is slowly overcoming me...
The cable was not to blame, I tried another one without any improvement. I also tried other ports of my router...
I had the hardware wizard re-install the NIC. Indeed the "Media Type" is appearing only once now, but the 100 Mbps Full Duplex option is not available (see pic). What should I choose : Harwdare Default, AutoSelect, or another one ?
There is also one weird thing : everytime I go into the NIC adapter>properties, and EVEN IF I DON'T MAKE ANY CHANGE, I am asked for the Windows 98SE CD when I click "Ok" to exit the properties window (clicking "Cancel" doesn't have this result). It is looking for the file PROTMAN.DOS, which isn't even present on the CD, so I have to redirect it to C:\WINDOWS where the file already exists. Is this a normal behaviour ???
As soon as it will be possible, I will try the wireless card in this computer to see if the 3Com NIC adapter is the source of troubles or not. I will let you know.
Thanks for your help