Windows doesn't detect any cable for your Ethernet adapter, so as far as Windows is concerned your computer and the Dell Dimension are not connected.
Ehternet Adapter Local Area Connection
Media States --- Cable Disconnected
Is the two computers connected directly to each other with a RJ 45 cable or do you have hub/switch/router in between?
If they're directly connected, have you remembered to use a crossed RJ 45 cable? A regular one wont work.
(Oh and you should probably edit the IP address for your Dial-up connection out of your last post. It is a WAN IP so in theory someone could use it to hack you.)