For hobby flight applications, lithium polymer ("Lipo's") generally offers the best performance from a weight-to-power or weight-to-energy ratio. Compared with the theoretical energy density of most cells, Lipo's are quite close as they usually lack the steel casing of most other chemistries.
Their downside is a somewhat fussy nature about how they are charged (you must use a Lithium battery charger) and discharged (going below a certain voltage can severely damage the cells, turning them into a potential fire hazard when recharged), and their poor mechanical robustness, due to the aforementioned lack of a steel case.
An additional potential negative in using lithium batteries with a NiMH-designed circuit is that their voltage varies dramatically over their discharge curve, from up to 4.4 V when charging down to 3 V when fully discharged. To fall within a 7.2 to 12 V input, a 2 or 3-cell Lipo should work. They are usually charged at 4.3-4.4 V/cell, so a 3-cell's open-circuit voltage may be above 12 V. If the upper limit you cite is absolute, you would be stuck with a 2-cell Lipo, which will drift well below the lower limit.
So if you're looking for an upgrade from NiMH, Lipo is the way to go. Respect their hazards and they will provide much entertainment.
Let's get the easy one out of the way. AC batteries can never, and will never happen, because the chemical reaction that creates the voltage will only ever take electrons from one electrode and deposit them on the opposing one. To have AC instead of DC, we can use a device called an inverter to switch the DC back and forth at the correct frequency, and do some other cool things to make it a better source of AC power than just an ordinary square wave.
Now the terminal voltage of a battery...

Here is a simplified model of what is going on. When you pick up an unknown battery, you have no idea what it's internal resistance is; if the battery is really old or damaged, it could be very high. And since real voltage sources are nothing like the ideal one is this model, you do not know how much instantaneous power it is capable of outputting.
So let's say this battery is outputting exactly 12 volts, but it's internal resistance is really high. If you put an ordinary voltmeter across it's terminals, that voltmeter doesn't draw very much current, so it will register 12 volts. When you attempt to use the battery, the current across the resistance reduces the voltage at A significantly, and you can now see that the battery is not usable.
The same thing can happen to a battery that has very little charge left, even when it has normal internal resistance. When you attempt to draw a load on the battery, the voltage source simply falls in value, as the chemistry no longer has enough energy left to sustain it. In this case, it is typically more obvious, as the voltage will be abnormally low when read by a voltmeter, but the fact remains that it will show a voltage, and that voltage might look adequate, especially in a lead acid battery.
There are professional grade handheld testing units that can test a battery by forcing a very brief high current discharge and concurrently measuring voltage and current. These readings then allow you to directly measure internal resistance, and can also be used to estimate a value known in some industries as Cold Cranking Amps. You can use a similar method on a smaller scale to verify that a battery has at least some usable capacity left by putting a reasonable resistance in series with the battery and taking voltage across and current through it.
Best Answer
No
A "perfect" battery might work for this, but any real life battery is going to have big problems. It will work OK to begin with but the current is flowing through the battery in the wrong direction, i.e. backwards compared to when the battery is being used normally. This is the same as if the battery is being charged, so the battery will quickly become overcharged. The consequences vary depending on the amount of current and the battery chemistry but may include:
To drop 12V to 9V for a variable load, use a regulator. To get the most stable voltage, use a linear regulator. If efficiency is more important, look for a switching regulator. Regulators are very cheap - my usual supplier has a bag of 10 suitable 100mA regulators for sixty pence (less than a dollar US).