The noise is amplified by the gain of the amp and will indeed appear on the output. You multiply 40nV/√Hz by the square-root of the bandwidth of your circuit. Thus if you had a circuit with a gain of 10 and a 10KHz bandwidth, the output noise would be 10 ×√10kHz × 40nV = 40µV.
The noise has nothing to do with the frequency of any signals going through the circuit, only the circuit bandwidth frequency. And it does not reduce with frequency (although 1/f noise, which is a type of low frequency random noise also present in amplifiers, is inversely proportional to frequency).
Therefore, to minimize noise you want to add a filter to the circuit to limit the bandwidth to no greater than you need.