Measure average and instantaneous pulse power
1) How to calculate average power by having voltage and current waveforms of one oscillation period that looks like this?


Integrate current times voltage, as you have done.


  Won't this integration result in "negative power" somewhere between 5 us and 10 us and then again after 15 us? Applying this formula to waveforms that overshoot results in negative power.


Yes.


  Sine waves with purely resistive loads should result in zero power (hence, RMS?).


No, sine waves with purely resistive loads should not result in zero power, where did you get that idea?


  Shouldn't an RMS variation of this integration be used for average power measurement?


No. RMS is a shortcut for doing the integration.



Negative power means power is coming out of the load, into the source. Whether you add this to the total power, or subtract it from the total power, depends on what you're calculating for.