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I have a few theories on these measurements.
1) Gain and level can be found to have an equilibrium using a graphing calculator equation leading to better harmonics.
2) MAX Power can be calculated somehow, and I believe that an industry specific regarding this is +3DB. However I am completely unsure.
3) If anybody else has any theories in this regard of dialing in these basic equations using a ti-84 please comment, I only have these two theories.
How detailed of an answer are you looking for? Do you want to read the official standards that define these things? They are difficult to read and not free. Look for AES, EIA, ANSI, and IEC. I can give you specific standards numbers if you really want them.
Gain and level are related by a straight line on a graph, yes, if you plot them both with linear axes or both with logarithmic axes. If you plot gain in decibels vs level in volts then you won't get a straight line because decibels are logarithmic. I have no idea what you mean about better harmonics though.
RMS and MAX power can be measured in so many different ways that it's not possible to give you a single answer. These can also be defined by the standards I mention above but again, those are hard to read and not free.
It's like trying to define how long a phone's battery will last. If you want to claim your phone's battery lasts longer, then measure it when the screen is off the whole time. Or turn off the bluetooth radio. Or don't make phone calls. Or whatever. And you'll end up with a number that's really impressive but meaningless.
I might be overthinking your question.
RMS is the root-mean-square of the signal. RMS power is the mean-square of the signal divided by the nominal impedance (like 4Ω). The RMS power of a 10v (rms) signal into a 4Ω speaker would be:
10 * 10 / 4 = 25w
MAX/PEAK/MUSIC/DYNAMIC power is usually loosely defined as twice the RMS number. Some of this is because music is dynamic so your 100w amplifier is only delivering a fraction of that to your speaker over time. So your 25w speaker will easily handle music being played by a much larger amplifier so long as you're not clipping badly. The MAX number is sometimes used as a guide to what size amplifier you should pair a speaker with, but not always.
Gain is usually measured in decibels and is related to level by the equation:
Gain [decibels] = 20 * log10( new level / old level)
The gain of going from 5 volts to 10 volts would be: 20*log10(10/5) = +6dB (which is 4x the power)