It sounds like you may have a handle on your question already, but I would like to add a couple of points about NIMH vs Alkaline.
You don't say how many watts or how many amps or milliamps your megaphone will be demanding, but the NiMH have a much lower internal resistance and so they can provide a much higher current without dropping their voltage as much as an Alkaline will. At higher loads, a Nimh will provide more power than a Alkaline; at very low loads, an Alkaline will provide more power (a very very loose generalization, but for example, an Alkaline will last forever in a remote control while the NiMH will not last as long - of course there is also the self discharge of many of the NiMH).
Also, most NiMH have a NOMINAL volatage of 1.2, but fully charged they start out at closer to 1.4 (I have measured some NiMH fully charged at 1.5) volts, and as mentioned already, hold to the 1.2 volts for most of their discharge.
The Alkalines often start closer to 1.6 volts, but quickly lose voltage as they discharge so that their average voltage through their life is about 1.2 Volts! Of course this depends on what the cut off voltage of your Megaphone is.
Don't forget that most NiMH have a very high self discharge rate and can lose 10-20% of their capacity in the first day, and 1% of their capacity per day just sitting there and are essentially completely discharged in 3 months or less! There are newer NiMH that hold 85% of their capacity over a year.
Check out http://batteryuniversity.com/learn/article/Nickel_based_batteries for more info.
As far as your 4 options, option 2 sounds like the best one. Options 3 and 4 have been discussed by others as to why they offer such poor performance.
A 12 volt battery has 12 volts between its terminals. Voltage is always measured between two points - there is no "absolute" voltage.
If you have a 12 volt battery and an 11 volt battery, and connect the negative terminals, the voltage between the positive terminal of the 11 volt battery and the positive terminal of the 12 volt battery will be +1 volt, assuming the negative lead of your meter is on the positive terminal of the 11 volt battery.
If you connect the positive terminals instead, there will be a 1 volt difference between the negative terminals.
The term "Ground" is much misused in the electrical field. Occasionally it does mean a connection to the Earth, but most often it is just the point in a circuit that the designer chose to call "zero volts", and uses as a reference when measuring voltage elsewhere in the circuit.
Best Answer
There is one thing which is obvious once stated, but not until then.
Your phone tells you it has "37% Charge Remaining". How do you know that's accurate? It's probably not.
The software may be doing some estimating based on average current draw since it was fully charged, average time between charges, and of course the discharge characteristics for the specific battery. Then it presents you with its best guess.
Over time, it can build up a reasonably accurate profile for the battery and use that to improve the estimates. But it is usually an estimate.
In my experience of developing battery based systems (with smart batteries, dumb NiCad, and everything in between) the only times you are confident of the charge level are 100% and 0%.
Usually, a smart battery will let you know when it's fully charged, and with a dumb one you are probably doing some calculations with current and temperature. That takes care of the 100% case.
The 0% case is where the sneakiness comes in. Whatever the battery chemistry, there is often a distinctive pattern in the discharge curve as you approach voltage collapse. But allowing a battery to go into deep discharge is generally a "Bad Thing" (TM).
So firmware looks for that pattern and decides when the battery is at a virtual "0%". Then it shuts the system down so that there's enough residual charge in the battery to avoid deep discharge and, more importantly, a sudden loss of power. This allows a graceful shutdown.
If this seems a little unlikely, let your phone "run down" and shut itself off. Then turn it back on again. If the battery were truly at 0%, it could not boot and power up the screen to tell you it needed charging.
The 5% (or perhaps 10% depending on the precision of the measurements and the batteries tolerances) warning is also often somewhat artificial, again representing a point on the discharge curve when the firmware starts thinking "Going to shutdown soon".
Ironically, this is the level at which someone in marketing insists that you turn on that bright LED to tell the user they are about to run out of battery power.