Without a datasheet, nothing is certain. But Sanyo do seem to be masters of the art and tend to lead the field in LSD NiMH. Their claim for a 1500/2000 cycle lifetime (eneloop / eneloop lite) and 75% retention at 3 years is superior to anything else I've seen so far. They also claim operation at down to -20 °C. No curve shown but liable to be acceptable given -10 °C curve.
So, I'd expect that you may well get inferior performance from other brands — especially from those who provide no datasheets.
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.
Best Answer
It does! The capacity of a battery is correctly measured in watt hours (or equivalently, joules), not amp hours. A rough approximation of a battery's capacity in watt hours is its rating in amp hours multiplied by its nominal voltage.
Putting two 1V 1AH batteries in series results in a 2V 1AH battery - which has twice the nominal capacity. If you were to use the battery as an input to a buck regulator (known as a BEC in RC hobbyist terminology), you'd be able to draw 1V with 2AH, before accounting for losses in the voltage regulator.