1) Turn signals haven't used thermal relays for quite some time now. Modern systems tend to use flasher relays that are either capacitor based or transistor based.
2) With the old thermal relays, the more bulbs you have (and hence, the more current flow you have), the faster it flashes. The old thermal strips would either flash very slowly or stick "on" when a bulb was out.
3) New flashers don't "overcome" the fast flash with a bulb out at all. In fact, they rely on that as an indicator for the driver and often explicitly state such in the owner's manual.
The impedance is a combination of resistance, inductance, and capacitance.
impedance = resistance + j(inductive reactance - capacitive reactance)
Inductance and capacitance are usually not a big part of DC impedance because inductance is related to current fluctuation and capacitance is related to voltage fluctuation. Since AC is all about constantly changing voltage and current, both inductors and capacitors can put a hard limit on how much current can flow through an AC circuit. DC doesn't change direction the way AC does. In DC, inductors prevent rapid current change and capacitors prevent rapid voltage change, but neither puts a hard limit on how much current can flow.
Zenmervolt is right about them using capacitors and transistors. A transistor is a voltage controlled switch, and capacitors fight against changes in voltage. When used together, a capacitor controls the voltage in part of a circuit, and that voltage going above a certain level allows current to flow through a transistor. By charging and discharging a capacitor, it allows a transistor circuit to open and close. If the speed of this cycling changes, it means something in that circuit changed. Maybe the alternator and battery voltage is too high (not likely), or the resistance of the circuit changed because one of the lights just burned out