Not sure about getting more AH- I think you mean the Watt- Hours are constant- when you reduce the voltage, you get more amps, minus the loss in the converter. Just remember that the unit itself has a converter, from 12V (13.6 down to 9 or so) to 5 volts or less, whatever the board runs on. You are really not saving power by reducing the input voltage to 12V- you are just giving the internal DC/DC converter a lower voltage to work with, and more amp- it is really the same power from the battery (minus losses)
A battery has so much energy and that is volts X amps X hours, and if you convert down to 12 you have not made more power, you have just traded volts for amps, and lost a little in the conversion.
Exactly.
It's all Ohms law.
Batteries have only so much stored energy (as you said).
I have done some load testing with tool batteries and buck converters.
This is the tester that I have used for years, very accurate.
http://www.westmountainradio.com/product_info.php?products_id=cba4.
Start with a battery (doesn't matter the type or voltage) and set the buck converter output to say 12.5 volts with "X" amp load and check battery run time.
Same test with the buck converter output set to say 14 volts with the same "X" amp load and the run times will be for all intents the same, they may vary a bit but that could be battery recharging etc.
Another thing to remember is that most lithium-ion batteries have a minimum/maximum voltage that depends on the chemistry of the cell. Some batteries are now 4.35 volts per cell, times 5 cells (18/20 tool battery) equals 21.75 volts.
At the other end is the lowest battery voltage which it should never be "loaded" below. There again that depends on the chemistry as well. Generally the low voltage is about (depends on battery) 3 volts, so 3 volts x 5 cells equals 15 volts.
With the above tester the shut off voltage can be set so the tests are equal.
Like you said the load (fishing electronics) is only using "X" amps.
And as always your mileage may vary.