I too learnt that story or Al and P in silicon, with electron and holes conduction, and didn't feel it so senseless. At least, it could explain much of what I experimented. And since this very theory is applied with success to produce chips, there must be some truth in it.
I haven't studied ZnO, but if comparing with GaAs, I suppose the effect of a dopant depends on what atom it replaces... Al replacing Zn would bring one more electron, making it N-type. Within the crystal, Al would become Al3+, replacing Zn2+. And if Cu has a valence of 2, it won't dope anything by replacing Zn. As well, a dopant level must be near enough to the conduction or the valence band, or it will be a recombination centre instead of a dopant.
A dopant tends to replace the bigger ion in the crystal, which must be Zn2+ rather than O2-. But in a thin film, which uses to be polycrystalline or amorphous, these simple reasons that work for single-crystals can be very wrong, as grain joints determine most properties and suck impurities.