I'm a little confused on a few things, first let me make sure I'm looking at the correct router:
Linksys wireless G
For clarity purposes:
CompA = new computer
CompB = other wired computer
With compb, you said you got a normal ip when plugged into the router. What is the normal ip? It should be 192.168.1.x The value of x will depend on what the dhcp server gives it. The 169 ip address you received when plugged directly into the modem is not the modem ip address but the ip of that computer. When it cannot obtain an ip address from DHCP, it obtains a 169.254.x.x ip address. This is a private ip address that cannot be used to access the internet. If that comes up try the follow commands at a command prompt:
ipconfig /release
ipconfig /renew
and see what ip that gives you.
Assuming that is the correct router I listed above, the ip of the router will be 192.168.1.1 When you try to ping the router, ping that ip.
For CompA, when you are directly connected to the modem and get an ip of all 0's, Do a ipconfig /renew at the command prompt. That should give you an ip from dhcp. If it hangs there after running the command, check to see the driver for your nic is installed.
This is a link to the user guide for your router. Run through the steps in chapter 5 and verify the basic settings are correct.
users guide