Hi Everyone,
Thank you for all your help. In the end, something was preventing the
drivers from loading for the NIC card. I downloaded the drivers for this card and installed them and it connected to the Internet just fine.
I did a thorough scan with MalwareBytes and it found nothing.
Chev65, I think if you look around you will discover that layered security is a common and recommended approach. The security applications I mentioned are not all AV. One is a firewall, one is AV, one is malware protection and one is behavior-based protection. These are all running fine in harmony on my new Windows 7 system.