Calibrated X, Y, Z, and Ex stages by running them back and forth while turning their potentiometers down until they stopped moving, then slightly up again. This cut down the idle heating quite a lot, to where it doesn't appear to be an issue anymore.
Trying to calibrate the extruder motor this way I had to heat the extruder first, which required reading the temperature sensor. Must remember to not set pinmode for analog pins! And that the yellow wire is for +5V, while the blue one is the actual measurement.
The extruder stepper motor driver behaves oddly -- where the other ones run nicely with the AccelStepper library, that one only moves when changing direction. Turns out I have swapped Dir and Step in the shield. Updated old pin description.
700 seems to work as a starting extruder temperature.
The Z stage wobbled back and forth when changing direction. Need to loosen the roller skate bearings.
The Ex stepper motor driver still gets too warm after a while of idling, but the Y driver is fine now. Hopefully the Ex driver will be ok since it'll be moving most of the time. Some stress-testing of the Ex driver alone shows that it can keep going for quite a while (minutes, at least) without getting overly warm.
After loosening the roller skate bearings, the Z stage wobbling persists. It is probably caused by the slight movement of the rod-bearing roller skate bearing, which should be held in place with a properly cut piece of cardboard. However, since the Z stage moves so slowly I will consider that a minor problem for now.
While checking if the extruder head really comes just down to the printing surface, it turns out that the back edge of the plate holding the extruder is still hitting the frame, holding the whole extruder mount about a centimeter above the surface. I guess I need to slice off a few millimeters more.
But apart from that, everything seems to work smoothly:) So for next time, I will look into the higher-level software and how to make it work for my parameters.