It WAS the monitor

I took both the monitor and the tower in. It was the monitor. The technician was very nice. When I explained, he said he'd try testing them both. To his surprise, the monitor behaved as I said it would. I have a new monitor. The fan has been replaced on the tower and all is right with world.
However, I now know about backing up to external sources places (thanks Ros) and various other little lessons such as not having things saved in bookmarks means I spend less time faffing on the Internet...

The wip is calling...