AVG's support said that they didn't get the diagnosis file from the first attempt to send it to them and would I please start over? By that time, I'd already used System Restore to roll the system back to the 7.5 installation, so I had to indeed start over.
Just to be certain that there wasn't anything else in the mix, I did these steps:
With AVG 7.5 installed:
1. Can I launch my Outlook e-mail (which does connect to my work account) without VPN running? Yes, after I fixed the address to which it has to connect. It
2. Can I connect to my company's VPN? Yes, it properly connects to my company's network.
3. Set a System Restore point, just in case.
Now I ran the entire AVG 8.5 installation process again, downloading a fresh copy of the installation file just to be certain. After it installed, ran, and restarted, I tried these things:
1. Can I launch my Outlook e-mail without VPN running? Yes, it connects.
2. Can I connect to my company's VPN. No. It will not connect, saying the host is not responding.
3. I tried all the trouble-shooting steps in AVG's tech support e-mail, and none of them worked.
4. I ran their data-collection program (which gathers more in-depth information), recreated the error again, and this time it appears to have managed to send AVG's tech support the necessary e-mail.
As much as I want to get that machine back to being a proper backup for my main computer, I'm going to hold off restoring the system back to 7.5 and give AVG's tech support a chance to try and digest this problem and maybe suggest a solution. I fear the answer is going to be, "It's not our fault; it's Cisco's badly-behaved VPN client, and we can't do anything about it."