1). Exactly what is being meant with the term "head room?"
Head room is a vague term for added dynamic range. It can apply to speakers, amplifiers, or their combined performance.
All speakers have a range of loudness, from the quietest to the loudest, over which they can produce sound without deviating (too much) from the amplified electronic signal. All amps also have a dynamic power range. When a speaker is underpowered by an amp, the amp runs out of ability to deliver power before the speaker runs out of ability to get louder. The amp clips while the speaker can still get louder without distorting or running into thermal compression.
In that sense, a more powerful amp would generate more "head room".
All this is a vague generalization. Limits to dynamic range or head room depends on both the specifics of the amp and the speaker.
It pays to remember that power is a an estimate, not a direct measurement. It is calculated based on direct measurements of voltage and current. Of course, in speakers, current varies with the impedance, which varies with the frequency of the audio signal. It isn't a constant. Power also varies significantly with time. Some sounds, which demand high power, are very short, while others last longer.
How much headroom would I gain from doing this? Gene thinks it could be of some benefit, but I'm still waiting to hear an explanation.
You only gain head room if the original amp lacked enough juice to drive the speakers over their full dynamic range. If the original amp was powerful enough, then you gain nothing by adding more. Again, this is hard to accurately predict, so as a result, most people add more than enough power, just to be sure.
Edit:
Again, this is hard to accurately predict, so as a result, most people (who fret over head room) add more than enough power, just to be sure.
In reality, MOST people don't care.