How can I extract the rms noise from the measurement's over all noise?
Should I short the scope inputs


Yes but make sure you are still grounded to where you want to measure the noise from your amplifier - there may be some common-mode artefacts that bump up the noise even when the scope inputs are shorted to each other and, if the scope inputs were not grounded (but floating) you might not capture them (if any).


  Does such a method neglect common mode related noise


If you keep the ground, grounded it should be more accurate. The formula is correct by the way and take several measurements to ensure agreement.