Yes, you can use a common power supply for all five amplifiers. The supply must be capable of providing enough current for the peak demand of all five amplifiers at once (e.g., 5 × 125 mA = 625 mA). For example, the NT101A/NT102A power supply shown in the datasheet has a rating of 650 mA, which would be plenty.
And yes, it does not need to be a regulated supply, as long as it always produces a voltage in the specified range.
Those input power and voltages are rated input power and voltages.
For example you can drive electric motors over rated power but they will get too hot and eventually break. There are also electric motor duty classifications for industry.
See this for short introduction:
In practice this means that one same machine could be used for two different applications.
For constant non-stop usage rated power can be e.g 100 W but then for cyclic usage where motor stops e.g for 5 minutes and then drives for 2 minutes the rated power can be e.g. 130 W.
This was just an illustrative example from industry machines but I have not checked how big difference in power output there actually is between these two types.
Back to this case:
Peltier element's rated input power in this case is that around 96 W. You can also use it with lower power. For example you could attach a sensor system that measures temperature of cooled object and then the input power is adjusted by control circuit to adjust for example voltage given to the element. Since peltier is a semiconductor device it is likely more prone to break with over-rated power even for short times. I do not recommend trying that.
The rated numbers for that element can be based on theory and then it is tested properly to be sure that it works under that load for long time enough to be sold for consumers.
Also shortly about fundamental theory:
- Voltage (potential difference between two planes, nodes... etc.) produces electrical current -> resistance limits current -> power is consumed to that resistance to get over it. Refer to Kirchhoff 1st and 2nd and Joule's law.
In practice you can buy cheap multimeter, small battery and a small resistive lamp, couple of resistors and see with measurements when you change resistance in that circuit and see in practice in brightness of the lamp. This is brilliant way of getting started in practice.
Remember to stay safe while measuring and do not measure any high power device voltages or currents if YOU are not familiar with the theory of electric laws! Small 9 V alkaline batteries are safe enough but things get much more dangerous even with 12 V car batteries if you don't know what you are doing!
Just as an example of where the manual doesn't appear to match the product label, some switch mode power supplies may be labelled 100 volts to 230 volts but UL testing and CSA testing will be 15% above and below those values and, the manual is quite at liberty to give the fuller range ie 85 volts to 264 volts. I've seen this just recently and had it confirmed by the reputable supplier.