Why must the order of a Band-Pass and Notch filter always be even?
In principle there is no reason why the filter order of a general bandpass or bandstop filter must be even. Such a restriction is a consequence of a specific design procedure. In classic IIR filter design (Butterworth, Chebyshev, Cauer) you start with an analog prototype lowpass filter. Bandpass or bandstop filters are then obtained by a frequency transformation. And it is this frequency transformation that doubles the order of the prototype lowpass filter, hence the even filter order for that specific design method. Note that the bilinear transform has nothing to do with that restriction on the filter order.
Of course, there are specific designs for which an odd filter order doesn't make much sense. E.g., for a notch filter (with a notch frequency greater than zero) you want exactly one zero at the notch frequency plus its mirror image at the negative notch frequency. So for each notch frequency you get two zeros, and, consequently, two poles.