Choosing proper stepper motor, motor driver and power supply combination
Well, it seems that you are looking for some general guidance, instead of presenting a real electronic engineering question. Most of folks here don't like this kind of question. But, anyways, I can say something about your doubts...
1 I am planning to feed the motor driver with a 12V supply. Will the motor driver be sufficient for the stepper motor I will use? (I don't know how much current the stepper motors can draw at max with this voltage, that's why I cannot make sure)
It's impossible to know for sure, since you didn't described your application. For most general uses, A4988 should be sufficient for this kind of stepper motor.
Steppers controlled by chopper drivers are somewhat weird: the applied voltage is not so much important as the current. For most general uses, 12 V are ok.
2 Generally for this motor driver it is suggested to use an electrolytic capacitor between VMOT and GND. They say if it is not done spikes can be a problem. Does it mean that it will damage the driver or can it have other impacts? And does it have to be an an electrolytic one? The electrolytic ones I was able to find was SMD and I don't know how to fit them.
Yes, spikes can damage the driver. Or something else can happen, and even if the driver is not damaged, the circuit may not work as desired. A4988 datasheet specifies 100 uF, so, follow what the datasheet says. Yes, it has to be an electrolityc capacitor. There are electrolictical SMD capacitors, aluminium-can type.
3 - I have found a CNC shield for Arduino Uno. But the shield has sockets for only 4 drivers. It seems that I will need to run 4 Arduinos at the same time. Do I have any other option? If not is it possible to run 4 of them properly (as my laptop does not have 4 different USB ports I will need to use a splitter)?
Simply don't use Arduino Uno, it is somewhat slow and has few outputs. There are a plethora of microcontrollers available that could fit your project better, whatever your project is (you didn't described it). 15 steppers in one circuit is not a trivial solution and probably you will not find a ready-made board to install those drivers... This is also a somewhat crytical design to fit in breadboards. Probably it would need a custom PCB.
4 I am planning to use a 12V power supply. How much total current should I expect at max? I need to choose the current capacity of the power supply.
As I said about question 1, it's impossible to know, since you didn't described your application. If the motor draws 1.5 A per phase, and each motor has two phases, there will be a theoretical maximum of 3A per motor, 45A total for 15 motors.
Maybe the total current drawing will be lower, but this it is not a reason to save a few bucks in a 30A or 40A supply. The price difference is insignificant (knowing that you will spend a lot in motors and drivers), so, go for a 50A or even a 60A or 70A 12V power supply.