In this case, you are being led astray by what you don't know. A speaker coil has a (more or less) constant impedance because it drives a cone against air resistance. In your case, with no speaker attached, it behaves as an inductor, not a resistor. From the numbers you provide, it will have an inductance of ~ 5 milliHenrys (.005 H). For an inductor, the current (and the magnetic field) will go down as frequency goes up. The relevant equation deals with impedance, not resistance, represented by Z.
Zl = 2 x pi x L x f
and this will equal 8 ohms at f = 250 Hz (about). So, for frequencies below about 250 Hz, the total impedance (sqrt(Zl^2 + R^2)) will increase with frequency, until at 250 Hz your current amplitude will be about 70% of its DC value. Further increases in frequency will continue to increase impedance and current will drop further as you increase the frequency. You probably don't need to hear about phase shifts.
The effect of all of this is to reduce power dissipation in your coil as frequency increases. But particularly at low frequencies you do need to keep a very close watch on your temperatures. Dissipating 800 watts in a coil the size you're considering stands a pretty good chance of melting the interior layers. It certainly nay cause the bobbin to melt or catch fire.
Without doing any real engineering, those in line volume control potentiometers might actually not be such a big deal, provided you insert them in the correct location.
See, the input to most amplifiers is high impedance - meaning, essentially, that it won't draw much current. Using a potentiometer to attenuate the line before the amplifier is a quick and dirty way to get things done, but as far as waste goes, losses around the potentiometer would be fairly minimal.
For several reasons, you certainly do not want to use this hack between a power amp and the speakers. That would be quite wasteful indeed.
When those amps are cranked up, that hum is simply the amplifier doing its job on a signal that probably came from a mildly noisy component or connection. Everything has resistance, and all resistance experiences thermal noise. It could also be power supply noise or a number of other things, but it can only be removed by careful design. If you use the inline volume control, there will now be an additional source of noise that may or may not be tolerable.
For other options, I'm honestly not sure of many other than buying or building a low distortion power amplifier to serve as the input stage to your last power stage. Fortunately, the kind of small one you would need for an intermediate stage can be fairly cheap, since all the heavy lifting is done further down the line.
Best Answer
Analog audio
Sound is transmitted through the air as variations in pressure. When a guitar string vibrates, a drum skin is hit or the vocal chords vibrate the air pressure oscillates a little around atmospheric pressure causing alternating compression and rarifaction of the air. A microphone detects these air vibrations and converts them to electrical signals. With no sound the microphone gives out 0 V. When it picks up a sound it gives out an alternating voltage going positive and negative with the air pressure. The electrical signal is an "analogue" (as in analogy) of the sound signal, hence "analog electronics" and "analog amplifier".
Dual-rail power supply
The alternating nature of the analog signal poses a problem for the electronics designer. Since the audio signal alternates around 0 V the amplifier circuits must too. To allow this a dual supply is required.
simulate this circuit – Schematic created using CircuitLab
Figure 1. This amplifier has a symmetric ±12 V supply. Note that there are no capacitors in the signal path.
If the The output can swing, say, to within 2 V of the supply voltage rails then the output could go from -10 V to +10 V. The instantaneous peak power would be \$ P = \frac {V^2}{R} = \frac {10^2}{8} = 12.5 \ \text W \$.
Note also that this amplifier has no capacitors in the signal path. That means that it can amplify "right down to DC" or, more relevant for audio, there is no fall-off in amplification at bass frequencies.
Single-rail power supply
There are many cases where only a single supply rail is available. The 12 V electrical supply of a car is an obvious one. The circuit of Figure 1 would not be able to handle negative signals and would distort terribly. To get around this we introduce a DC offset into the audio signal while it runs through our 12 V electronics. Typically this is about half of the supply voltage or 6 V for the 12 V system.
simulate this circuit
Figure 2. The amplifier modified for a single supply rail.
But we've lost something! If the amplifier can still swing to within 2 V of the supply voltage rails then the output could go from 2 V to +10 V - but that's 6 V ± 4 V and so the peak voltage the speaker will see is 4 V. The instantaneous peak power would be \$ P = \frac {V^2}{R} = \frac {4^2}{8} = 2 \ \text W \$.
One way to improve the situation here is to lower the speaker resistance. If we used a 4 Ω speaker we could get 4 W from this amplifier (provided it can handle the increased current).
Bridged mode
simulate this circuit
Figure 3. A simplified version of a pair of amplifiers in bridge-mode. Note that one amplifier is non-inverting and the other is inverting.
Main points:
I think I've covered that.
More power. Louder.
From the comments:
Ears are the best devices for this. If you carefully press the speaker cone from the front or rear it will deflect and will return to the centre position due to the suspension arrangement which acts as a spring. When you overdrive your speakers you will approach the limits of the suspension where it no longer performs linearly or even hits a hard stop such as when the voice-coil hits the bottom of the slot in the magnet.
Figure 4. A loudspeaker cut-away. Source: Wikimedia commons.
Over-driving as described above will result in audible distortion which generates harmonics (multiples) of the frequencies present in the sound. This will make the output sound harsh and this effect is used by guitarists either by deliberately overdriving their speakers or using guitar effects units to create the effect.
High quality speakers may have tweeters for the high frequency signals. As the harmonics may be above the limits of human hearing the distortion may not be noticable so caution is urged.
Of course, you can read the datasheet for the speakers and calculate the maximum signal levels that should be applied.
If you wish to study further you should be able to find some MP3 sinewaves of various frequencies to play through your system. Alternatively a signal generator app for your phone would allow you to do all sorts of frequencies, sweeps, white noise, etc, and a spectrum analyzer app would allow you to check the response from your speakers.