Remove a frequency and all its multiple (harmonics)
The choice of algorithm depends on your application scenario - i.e. there is no best solution per se. If your focus is on minimum computational requirements, a comb filter will probably be optimum. If you want optimum sound quality, a notch filter based approach could be a better choice.

To gain an overview about advantages and disadvantages of the different methods, maybe our article is of interest. We compared


comb filters
subband comb filters (only processing the lowpass frequencies) and
notch filters


in terms of disturbance reduction capabilities and degradation of the desired signal.

Answers to Your Questions


Yes, each notch filter creates unwanted side-effects. On the one hand, you will get audible ringing artifacts - especially noticeable for narrow bandwidths. On the other hand, higher bandwidths suffer less from audible ringing but bring about increased degradation of the desired signal.
Comb filters remove evenly-spaced harmonics in one step by placing notches that cover the full frequency range. As a consequence, you have no control about how many teeth they have as the notches reach up to the Nyquist frequency. If you want to have control about the affected frequency range, you can split the frequency range into high and low bands, process only one with a comb filter and add both bands together.
Except for the mentioned subband comb filter, none that I know of.