So, I finished my (rat's nest) wiring, and there doesn't appear to be any short circuits when I turn it on.
One small change - I wired all the momentary push buttons (4 timer switches, enlarger switch, and encoder switch) together in a voltage divider. When I tested it on the Serial Monitor, the values were all distinguishable which is great. All the resistors are the same 4.7k ones, so that's easy. Some tutorials online had different resistor values, probably to make sure the increments are linear rather than logarithmic.
The Uno I am using is a clone because I'm cheap, but it also has a very handy feature where the pins have solderable holes next to the female connectors, so I used them. I also had to desolder one segment of the connectors in order to squash everything in.
Next step is to start programming in earnest, the display is going to take a bit of thinking about because in the end I just soldered the anode wires to any old pin on the Arduino.