Here is some more data, in case it helps somebody. I did a fresh install of a VM with Windows Server 2008 R2, but the same issue!
I contacted JetBrains support, and they requested that I start Visual Studio with these switches: "devenv /ReSharper.LogFile C:\resharper_log.txt /ReSharper.LogLevel Verbose"
to collect Resharper log data. When I did, the machine lost network (as always), and it took several hours to come back. The log file was 3.7 GB(!) with millions of this line:
Thread:25: [ULCCC] Received from=x.x.x.x:63300, windowUserName=joe userName=Joe Doe
So it appears to basically DOS itself somehow.
I heard back from JetBrains and they asked me to try to change the license to Eval (instead of registered), and it handled the issue. I am waiting to hear back on a more permanent solution. So it is basically a flaw in their license check mechanism.