I'm at a huge loss to understand what is going on here. Normally what one sees when booting Knoppix on a computer attached to a router is that during the boot process Knoppix does a DCP discovery with the router and gets assigned a local IP address in the same IP range as the router, and has the broadcast address, Domain Name Sever and network mask set. From there they can access the Internet as soon as the boot process finsihes, no setup at all needed.

The thing is, you show that your system (at least one of them) is assigned 192.168.1.101, the broadcast address and the network mask are right. This isn't something that Knoppix ever does on it's own, this information must be coming from the router by DHCP. I'm not at all clear on why you couldn't access the Internet at that point, although we had seem some setup issues that deal with DNS servers that apparently don't treat IP6 properly. But then you do another DHCP setup by running netcardconfig and somehow the system no longer can recognize it's ethernet connection. That's a huge mystery to me and I can't make any sense of it.

Nor can I understand the ping to the router at 192.168.0.1 failing. I had expected the ping to knoppix.net to fail and the ping to the IP address here to work, but I didn't expect there would be any problem pinging the router, I just put that in because it's a logical first step in testing. Once that ping failed the others were sure to, but why it failed is strange.

I'm guessing that you tried the pings after trying the manual netcardconfig. And at that point, for some strange reason, the system no longer thinks it has an ethernet interface. The first test I would suggest is, if I'm correct above, trying the ping to the router right after rebooting, before trying a netcardconfig. You can do an ifconfig first, and that should confirm that the eth0 interface is still functional and that the computer has been assigned an IP address from the Linksys, but don't force a DHCP setup before doing the ping.

I also would suggest making sure that you try all of this from both systems and reporting the result from each.

As I said before, I would also check all of the settings of the router to be sure that there were no "security" features enabled that might muck you up, such as MAC filtering or IP address filtering.

We have on more than one occasion seen a router get into a strange state and do very unexpected things. I can't quite explain the eth0 vanishing from your ifconfig if that's what is happening, but it would be wise to try this:
  • Power down all computers, leave the CD in the disk drive.
    Cycle power on the Linksys, give it a minute to fully restart.
    Boot a computer directly into Knoppix without lettiing Windows boot first.
    Try the ping test again.

And just to be sure, we are talking about Knoppix running from CD or DVD, right? You're not one of those who thinks it can be installed and then fails to mention that, are you?