Ping is easy. Goto Run -> type CMD to open a command window -> type PING [IP ADDRESS] or conical name: PING
www.yahoo.comIt will give 4 attempts to 'ping' the address, returning a message of success or failure. Success will come with how long each communication took in milliseconds.
Scan the computer with Spybot and Adaware along with AVG. Seems they all have their own set of viruses and programs that they see. HijackThis is also a decent program that will show you inconsistencies in the Registry and hidden installed programs. It won't remove programs, just delete lines in the registry. But you can make note of them and remove them yourself.
Try winsockfix if you have issues connecting to the Internet. It resets all the protocols to factory default. It may clear up the proxy issue too.
In Windows 7, try adding the name of the XP computer to the user list. Try mapping a network drive too. Don't browse for it, just type the name of the computer followed by the name of the shared folder. \\COMPUTER NAME\FOLDER Sometimes specifying the computer name will kick it into gear.
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If you connect computers directly you need to use a cross cat5, not a straight.

There are 4 pairs of wires in a cat5:
Orange
Orange/White
Green
Green/White
Blue
Blue/White
Brown
Brown/White
In a straight wire, the sequence is (both ends) in an RJ45 connector: OW O GW B BW G BrW Br
Crossed: GW G OW B BW O BrW Br - OW O GW B BW G BrW Br
Cat5 is also called a patch cable, that term may be more familiar to you, but a patch cable can also refer to the cable that connects a phone to the wall jack. So a cat5 (category 5) is more specific in terms of shielding and number of pairs. Lots more info on this subject out there, but this is enough to get you through most situations.
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If you upgrade the XP machine to Win7 make sure it have plenty of RAM. At least 2GB if not more. Too many XP machines out there only have 256MB of RAM. But I agree with your assessment that a new computer made specifically for Win7 is the ideal.