Effect of DC component on the whole signal - comparison between normalised and non normalised
As I understand from the description,you are not just using the DC information but also information at other frequencies.

If this is the case: 

Generally,Normalizing a signal means subtracting the mean value of the signal from the data.

Y = Y - mean(Y) .......(1)


Fourier transform is a linear transform.So,any constant scaling proportionately scales up the transform.

Y = Y/max(Y)     .... (2)


If (2) is performed then the transform is scaled by max(Y) but the scaling varies with the data and this can be clearly seen in the 2nd plot of your description.