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Q1: I wouldn't say all systems output zero when power is off, but I think that's true of many systems
Q2: Generally, systems that do this are unstable. So, take any unstable system, and you will find it has an impulse response that blows up.
For analog systems, storage devices (caps and inductors) act like memory and create an IIR response typically. If you dont' have storage elements (i.e. resistors only), then you will get the FIR response. The tricky case here is delay lines. This is memory and storage that gives FIR (unless you put it in a feedback loop). Really, everything is IIR because you can never make a perfect delay line or a resistor without stray inductance and capacitance.