This has been happening on long-installed machines for quite a while. But in the last week, four different sets of Windows 7 64-bit install media, over seven different reloads, have had Windows Update get stuck, or freeze, while searching for Windows updates, eating 100% of one CPU core. Several changes in reload pattern were attempted, some of which included WSUSoffline, without success. Profoundly expert help provided the following. Before deliberately attempting Windows Update, we are to install these, in this order:

1. Servicing stack update for Windows 7 SP1 and Windows Server 2008 R2 SP1: September 20, 2016:
   https://support.microsoft.com/en-us/kb/3177467

2. July 2016 update rollup for Windows 7 SP1 and Windows Server 2008 R2 SP1:
   https://support.microsoft.com/en-us/kb/3172605

3. September 2016 update rollup for Windows 7 SP1 and Windows Server 2008 R2 SP1:
   https://support.microsoft.com/en-us/kb/3185278

Windows Update may get stuck again, searching for new updates, the first time any of the above is attempted. If this occurs, each time, we do this:

1. In TASKMGR, terminate process wusa.exe
2. Restart service wuauserv
3. Try the install again.

The second time we do each install (after the termination and service restart), the popup for searching for Windows updates comes up for 3-10 seconds, and then we are asked whether we want to install. We then say yes, and it installs. Then we move to the next one of the three!