Does delta-sigma ADC also reduce Gaussian noise on input signal to ADC or just quantization noise?
No!  The ADC (delta sigma or not) can not reduce the uncertainty in the input.  It sounds to me that your friend has not made up a real signal flow diagram and then formed equations.  The answer to your second question is affirmative.  In fact it's generically  true that putting as much gain, without introducing extra, in the front end is a good idea.  But... think long and hard about input overloads distorting the signal and possibly overloading the amplifier circuit causing distortion and such; either in the front end amplifier or during subsequent signal processing.  In the "good"  old days overloading due to the input could cause numerous problems; including long term effects.  OTOH: detailed knowledge of the signal and noise spectral (or even time transient) characteristics can reveal ways to optimize your S/N .