Electronic – Why rechargeable batteries use 1.2V

Using the 1.4V is fine - the batteries need to be charged at a slightly higher voltage than their "nominal" voltage marked on the case.
The 160mA is probably the rate it charges at - if there is no end of charge indication, then don't leave them in there too long (more than 12 hours or so if they are fully discharged)
It may change over to trickle charge mode when the battery is fully charged (hopefully if it's not a really nasty charger) but without knowing more about it (do you have model number?) it's best to be safe.

If you read the Wikipedia page on Ni-Mh batteries (I'm assuming yours are of this type) you can see the recommended charging voltage is between 1.4V-1.6V per cell. Also reading further on that page, some seem to think that trickle charging at C/10 (0.1C) for extended periods is not so bad (although others disagree, and it will shorten the life of your batteries anyway)
C/10 for your batteries is 1800/10 = 180mA, so your charger is supplying slightly less than 0.1C.

So I wouldn't worry about anything really bad happening, although I'd get a better charger when you can (unless you can confirm the charger switches to a much lower trickle when the battery is charged)

The time it takes to charge can be roughly calculated by your battery capacity / charge rate, so in your case:

1800mAh / 160mA = ~11.25 hours

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.