How do we compute power from current and voltage samples? Can we quantify its accuracy?
How representative is p[n] of the actual power being drawn, p(t)?


Potentially, not at all. For instance, let's say that the applied voltage is a pulse waveform with 1% duty cycle - that is, the voltage is zero for 99% of the time, and (let's say) 10 volts for 1% of the time. The sensor is a 1 ohm resistor. Finally, let's say that the signals are sampled once per voltage cycle, and always during the narrow 1%. It should be obvious that ignoring the actual waveform will produce an estimate which is 100 times too high.


  Why might someone (say an engineer who's in charge of designing the
  power subsystem) be interested in sampling p[n] at a higher frequency?


The previous section should answer that one pretty well.

What you need to do is become familiar with the concept of signal bandwidth. For a signal with a limited bandwidth, if you sample at least twice as fast as the bandwidth you can accurately recreate the signal. Anything less than this will potentially produce inaccurate results.


  Also, what can we say about p[n] if the samples weren't simultaneous
  (i.e., i[1] was measured 1.5 seconds from the start and v[1] was
  measured 1.525 seconds from the start)?


That is even worse. Depending on the details of the signal, the resulting calculation can bear no resemblance at all to reality.

EDIT - As an example, let's say that the signal is a square wave which has levels of +10 and -10 volts. The load is a 10 ohm resistor. The signal transitions from +10 to -10 at 1.510 seconds. Then the voltage acquired will be 10 volts, the current will be - 1.0 amps, and the computed power will be -10 watts. Just in case you're really new to electronics, this is simply impossible for a resistive system.

As Edgar Brown has commented, if you know enough about the system, the delay between the two measurements can somtimes be compensated for (if the sample rate meets the Nyquist Criterion - but not (in general) otherwise.