By adding wire between the inductive and capacitive portions of a short but resonant antenna (while adjusting things to keep the natural oscillation frequency the same), will that wire radiate stored energy as well as energy supplied by the transmitter?
That would be tricky, since the length of this wire would have non-zero length, and would be part of the antenna. Can another wire be added to an antenna, I suppose so. Does this somehow demonstrate the antenna resonating? Probably not any more than the antenna already does.
If a high Q resonant antenna is "hard keyed", say by a fast digital output, will the resonant impulse response radiate at that resonant frequency (and potentially harmonics) instead of just generating flat broadband noise?
It sure will. You don't even need a fast digital output to perform this experiment: simply an antenna analyzer will do. If you generate a plot of return loss over frequency, you will see some frequencies are absorbed by the antenna because that energy is radiated, while other frequencies are reflected by the antenna. What happens to the reflected energy depends on the source, if it's a 50 ohm source then all that energy is dissipated in the source impedance and feedline losses.
If, say, a transmitter is tuned to near a high SWR band edge, after the transmitter is keyed off will the antenna continue to resonate and thus radiate, but closer to its true resonant frequency ? e.g. will a high Q resonant antenna "chirp"?
Yes. Q is by definition proportional to the ratio of energy stored to energy dissipated per cycle. If the Q is non-zero, some of the energy must not be dissipated in one cycle. Since there is no other place for the energy to go, it must necessarily make another cycle, with each cycle having diminishing amplitude.
And lastly, if SWR is measured carefully, will the SWR minimum be at the antenna system's self resonant or natural frequency of oscillation? Or at some other slightly different frequency?
Not necessarily: the SWR is at a minimum where the magnitude of the reflection coefficient is at a minimum, which might occur at an impedance with a nonzero reactantance, which is by definition not the resonant frequency.
The difference is evident if the antenna impedance is plotted on a Smith chart, such as this example from A wideband printed dipole antenna with optimized tapered feeding balun for ISM and FWA bands:
The minimum SWR points are where the curves are closest to the center of the chart, whereas the resonant points are where the curves pass through the horizontal center line. In some cases (like the solid line on the right) they appear to be nearly the same. In others they appear to be very different.
Of course, a perfect match is exactly (50+0j) ohms, so any antenna that achieves a perfect match must necessarily do so at a point of resonance. But any antenna with a less than perfect minimum SWR has some potential for resonance and minimum SWR to be at difference frequencies.