Choosing emitter resistor
In a car ignition application, the IGBT would be switching the ignition coil current which would be expected to reach several Amps.
This current flows through the resistor as well so the voltage across the resistor gives a measure of the current flowing through the coil.
The trace from that resistor to the IC to the left of the resistor is probably used to sense the current flowing through the IGBT (and coil).
Since the resistor looks to be something like a 1210 size it's maximum dissipation will be limited to maybe 250mW. The maximum voltage across the resistor with say 5A peak current would probably be limited to 50-100mV  to avoid exceeding that value. If it was 50mV this would imply a resistor value of 10 milliohms.
A short across the coil would cause excessive current to flow through the resistor which could have been the reason for the failure.
Note the large number of vias just beside the resistor - they are probably to get good thermal (as well as electrical) conductivity to spread the heat dissipated by the resistor.