Won't the back EMF from the relays hurt the LEDs?
At face value it certainly appears that the LED's would be damaged by back EMF as generated by the relays on turn-off. However if you take a look at the datasheet for the ULN2803 you see it is designed specifically for driving inductive loads such as relays, and has integrated clamp diodes between each output and common. The common point is connected back to Vcc of the circuit, thus clamping any relay back EMF.

The rate of change of voltage (as the magnetic field collapses and a back EMF voltage is generated) will not be fast enough to overcome the reverse recovery time of the clamp diodes and other latency due to track impedance etc. If you had to put a scope directly across the LED, I suspect you would see a relatively fast rise in reverse voltage to -0.6V, with only a small overshoot to perhaps a volt or 2. Nowhere near the 7V breakdown voltage of the LED. The only caveat here would be for a very badly designed board (which is not the case here) where there are very long traces back from the relay to the clamp diodes in the IC. In that case the inductance of these tracks would impede the reverse recovery diodes ability to sufficiently clamp the voltage in time.