I just have several simple questions.

1) For the circuit below, is it possible to perform small signal analysis on the circuit since the input source is around 1-2mv ( in other words, I can short circuit all of the capacitors in the circuit)?
2) What's the purpose of adding a capacitor in between R2 and R1 and the input source? I understand that the other capacitors are used for removing ac noise in the power supply.
3) Is it really necessary to use a single power supply for the microphone pre amplifier? Can I use a dual rail supply to power the pre-amplifier

 

1)
Yes, you can perform small signal analysis by replacing all capacitor by short.

What's the purpose of adding a in between R2 and R1

This capacitor is needed because the op-amp is supply from a single power supply.
When you use BJT as a CE amplifier, you use a voltage divider to bias the active device somewhere in the "linear region".
In case of single supply op amp you have to do the same think. You need to bias the op amp somewhere in the middle of his "linear region".
This is a job for 47K input voltage divider. 47K resistors provide bias voltage equal to 6V.
Without the capacitor the output DC voltage will be equal to 10V (positive sat voltage).
But we have this capacitor in the circuit, so DC voltage gain is reduce to one. And the output voltage is equal to 6V.

and the input capacitor

Input capacitor is a coupling capacitor.

3)
Now, you can use dual rail supply also.

 

Jony130 beat me to the punch but here's my answer anyway--

Question 3 is related to the other two questions.

Since the AC voltage from the mic goes both plus and minus with respect to ground, the amp has to amplify plus and minus voltages. If the op amp has only a single supply then it can't amplify the negative portion unless you bias the input above ground. That is the purpose of the capacitors. The 47k resistors bias the input of the op amp at 1/2 the supply voltage so it can handle the negative portion of the input signal and generate an amplified negative going output. The capacitors block this DC bias but couples the AC voltages.

Thus the answer to your 3rd question is yes, you can power the op amp with dual supplies. Then you can eliminate (short) the three 10uF caps.

 

1)
Yes, you can perform small signal analysis by replacing all capacitor by short.

This capacitor is needed because the op-amp is supply from a single power supply.
When you use BJT as a CE amplifier, you use a voltage divider to bias the active device somewhere in the "linear region".
In case of single supply op amp you have to do the same think. You need to bias the op amp somewhere in the middle of his "linear region".
This is a job for 47K input voltage divider. 47K resistors provide bias voltage equal to 6V.
Without the capacitor the output DC voltage will be equal to 10V (positive sat voltage).
But we have this capacitor in the circuit, so DC voltage gain is reduce to one. And the output voltage is equal to 6V.
Input capacitor is a coupling capacitor.

3)
Now, you can use dual rail supply also.

Alrighty! So let me see if my analysis is correct.

When I perform a large signal analysis (open circuit all capacitors), then it becomes a voltage follower and outputs only 6 volts. But when I perform small signal analysis (short all capacitors), it becomes a non inverting amplifier with a gain of 23.

 

Alrighty! So let me see if my analysis is correct.

When I perform a large signal analysis (open circuit all capacitors), then it becomes a voltage follower and outputs only 6 volts.

Well yes, but I will say that you perform DC signal analysis when "open" all capacitors in the circuit.

But when I perform small signal analysis (short all capacitors), it becomes a non inverting amplifier with a gain of 23.

Yes, your analysis is correct.

 

Note that if you use a dual supply, the input bias network should be replaced by a single resistor returned to common. This resistor may not be necessary* if the input capacitor has been eliminated, and the microphone is a DC conductive type e.g. moving-coil.

If the microphone may be disconnected or has no DC path, a bias resistor is still needed. For a signal source having its own DC voltage such as a powered electret microphone, an input capacitor must also be retained - and its polarity may not be as shown in this diagram.

*Except to define a load resistance for the microphone, which may be advisable.