Things have been proceeding very quickly thanks to Gary and Becky's help. I now have one bit being displayed correctly by my Arduino onto my quad alphanumeric display.
One small glitch became apparent this morning though: the quad display uses 2 of the analog inputs on the Arduino and I wanted to have at least 5 bits in my demo. That's 7 analog pins, but my Arduino Uno only has 6. I considered upgrading to an Aduino Mega, but I've ordered an analog multiplexer instead. Hopefully it'll get here quickly. In the meantime I can add up to 3 more bits before I run out of direct analog connections.
I'll be breaking out the soldering iron tonight. I found a 9V power supply that I'd saved from a broken device, so I'll be replacing the battery in the 555 timer circuit with that, and I have to desolder the socket from the board it's on. Then I have to figure out which pin is which. Then I want to fix the 555 timer circuit and my little voltage shifting circuit onto some stripboard; they've been on plug-board way too long, and it's making me nervous :)
It's so much fun to have a tangle of wires an components on the table in front of me and have a good idea of what's going on :)