Does Apple or Qualcomm do product binning for their SoCs?
Do they purposefully build chips that are driven lower than they could


Yes and no. I can't speak for these specific examples brands, but in a sense all chips are designed that way. They are designed with variability in mind, designed such that all chips meet specifications. Circuit techniques are used to compensate for variability (and some other tricks, like playing with voltages). 

In addition, there are cases where redundant bits are used. For example, the CELL CPU in the Playstation 3 console was manufactured with 8 SPE units, however one was disabled during testing. This way, they could "afford" to have one core not work and still have a working chip. Note that this is the case on every CPU - even if all 8 SPEs are working. 

Similar things are sometimes done on lower levels, and a full discussion would take too much time to write and be outside of the scope of the question. But binning is not the only possible technique that can be used to improve yields, and only a very select number of applications actually support it. For example, if your chip has any part that has to meet a standard, say the WiFi specifications, it either meets it or fails to meet it. Nobody will be interested in a "slightly less specification meeting" version.