At what internal resistance is a battery considered dead?
High internal resistance doesn't mean the battery is 'dead', just that it cannot maintain the voltage at high current that it could when new. The highest acceptable internal resistance is entirely dependent on the application. 

Rather than throw old batteries away I reuse them in devices that draw less current. I am using 10 year old 'worn out' rc model heli Lipos in devices that draw less than 1A. Even after hundreds of cycles under extreme conditions (10-15 minute discharge time) they still have plenty of capacity at lower currents.

Old Nicad and NIMH batteries tend to suffer from capacity loss and high leakage as well as high resistance. When a battery has greatly reduced capacity and won't hold its charge or can't sustain low current drain then it is time to throw it away.

"Internal Resistance" and "impedance" refers to the DC and AC characteristics. Internal resistance is traditionally measured by applying a load and reading the voltage drop, whereas impedance is obtained by applying a small high frequency AC signal (eg. 1KHz sine wave) and measuring its attenuation. Impedance is generally lower because the battery acts like a capacitor at high frequencies.