How is a common mode signal generated in a differential amplifier?
The differential amplifier amplifies the difference between the signals in the +ve terminal and the negative terminal. And no they do not generate common mode signals but rather block them

That is, Vout = Adiff(Vplus - Vminus) . Adiff is the gain with which it amplifies and usually a differntial amplifier has a differntial gain of 30-45 Db.

When both the +ve terminal and the negative terminal of the amplifier is given the same voltage then since Vout is just a scalled version of the difference between the input in the 2 terminals, it becomes 0 . For example let Vcommon be the voltage at the -ve and +ve terminal , now when you do  Vout = Adiff(Vplus - Vminus)= Adiff(Vcommon - Vcommon) =0 . Since noise is common to both the -ve and +ve terminal of the diff amp it gets cancelled and thus differential amplifiers are less affected by common mode signals/noise. Though this is in theory in pratice every diff amp is slightly affected by the common mode signals/noise. They amplify the common mode signal in a very small amount and this is because of the internal mismatches in opamps internal transistor design. This is called the common mode gain of the differential amplifier. A good diff amp has a very low common mode gain.
CMRR whihc is the common mode rejection ratio tells exactly this on how good a diff amp reject the common mode signal.

The image below shows the differential amp configuration and the resistors acts as a means to control the gain Adiff.