I don't have a smartphone, but don't most of them allow use of the phone at the same time as the phone battery is charging from a mains-powered charger? If so, they already effectively have a battery eliminator!
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Whether or not it is SAFE to do that depends on the characteristics of the charger and the battery. An old-fashioned heavy-duty charger with nothing more than a bit of resistance wire as a crude current limiter could well drive excessive current through the battery and cause a...
I think you need to set up the main PWM frequency, then use code to jitter the register value (in OCR2A/B) which sets the main PWM duty cycle.
From what I read (I'm not familiar with Arduino), although the default 'Fast PWM' mode has only a few preset frequencies it is possible, as Kelly says...
Yes. Perfectly possible for a receiving transducer and if 'tx' is interpreted as 'having been transmitted'. If, on the other hand, you are referring to the emitted acoustic signal, then perhaps a 'bat detector' could be used to monitor that.