Zero-phase-shift filters must be symmetric about the origin
Consider the Inverse Fourier Transform as the impulse response of the filter. In the case of an ideal brick-wall filter with zero-phase, the impulse response would be a Sinc function centered about zero in time (given by the Inverse Fourier Transform).



This is non-causal (time domain response is greater than 0 for $t&lt;0$), but in order to be zero-phase as shown, the time domain response must be complex conjugate symmetric.  It need not be real; in order for the frequency response to be real, the time response real components must be even and the imaginary components must be odd (complex conjugate symmetric). 

But to visualize the above plot further and what "zero-phase" means, a delay in time is a linear phase in frequency, thus if we shift the impulse response to the right (as done prior to truncation for a causal solution), the magnitude response is unchanged but the phase in no longer "zero-phase". If we continued to see the next plot I have in this series, which shows the impulse response truncated by the dashed window in this plot below, we would then see that the magnitude response of the filter would also be distorted, but that is beyond the question being asked here.



I summarize the properties of odd and even functions as well as other universal Fourier Transform properties in the figure below. The time and frequency domains are interchangeable for these properties; for example, if a function is periodic in one domain, it is discrete in the other. If a function has only real components in one domain (zero-phase), then all real components in the other domain must be an even function about 0, and all imaginary components must be odd about zero (inverse sign). This results in complex conjugate symmetry. 

The relationship of the Hilbert Transform mentioned for the causal and anticausal functions applies to minimum phase systems.



What is interesting is how these tie together and why I choose to describe the above relationships in terms of even and odd functions rather than simply saying the other domain will be complex conjugate symmetric. This way we can see how a causal function in time (or a single sided function in frequency) is the sum of even and odd functions, and therefore MUST be complex in the other domain given that we need both real and imaginary components to create such even and odd functions.   

Regarding the OP’s second question: a notch filter usually implies a very narrow band of rejection to remove certain frequency components. For more details see this post here: Transfer function of second order notch filter