Why do characteristic impedances matter only when traces are longer than half a wavelength?
A 1/4 wavelength trace or shorter can also have a substantial effect. The usual rule of thumb I've heard and used is that you can probably neglect transmission line effects when the length is less than 1/10 or 1/20 wavelength.

For a simple example, say you terminate a 1/4 wavelength line with an open circuit and drive it with a single-frequency source. After the signal reflects back to the source (1/4 wavelength away), it will look to the source like it's driving a short circuit instead of an open. That's a pretty substantial effect.

For more usual situation in digital design, you design the line as 50 ohms, and terminate the line with 50 ohms, but the actual characteristic impedance of the line might vary in production between 45 and 55 ohms. You want to know how big an effect that will have on signal integrity.

If the line is long, the signal propagates to the end, and reflects back. Then it propagates back to the source (which might not be well-matched at all) and reflects again. And so on. This produces a voltage at the load with a substantial ring on each rising and falling edge. The time it takes for this ring to die out is longer if the trace is longer because it takes time for those reflections to propagate back and forth.

On the other hand, if the line is very short (less than 1/10 wavelength at the "critical frequency" related to the rise- and fall-time of the digital signals), these reflections will all happen within the time the rising or falling edge is still in progress, and won't produce very much ring (overshoot or undershoot) at the load. 

This is why you will often hear a rule of thumb that impedance control isn't needed when the trace length is a small fraction of the wavelength.