It is tempting to assume the origin of the hydrogen is water vapor from the comet, in which case, what process "breaks up" the water molecule (or perhaps ammonia molecule?)? Is the hydrogen actually hydrogen ions? If the hydrogen is hydrogen ions, then could the source actually be solar plasma brought by the solar wind and trapped by the comets magnetic field?
Yes, ultraviolet light photodissociates water to form hydrogen. Both $H$ and $H_2$ are formed. See The photodissociation of water in cometary atmospheres for the relative amounts and a full discussion.