Am I understanding correctly that there are no requirements for cases?
Correct. The requirement is for the complete product. Be aware that the requirement is not just for your product to pass the test once in a type test, but for every unit in use to meet the requirements.
According to the EE in question constructiong a PC case from anything but (mostly) metal will result in excessive EMI radiation for my purposes (audio) leading to stray signals. Furthermore he says that ventilation holes (number, diameter, and position) are calculated based on stray em wavelength. Is this the case?
That's basically correct. It is possible to design a circuit that can pass the requirements with no shielding provided by the case; but doing so requires either a fairly simple product or extreme care (and probably trade-offs in functionality) when designing it. Most digital products you see depend on the shielding provided by their cases to meet the requirements.
It's also correct that the ventilation holes in your case should be designed in part to maintain the shielding properties of your case. Each opening in the case has the potential to act as a slot antenna, which will radiate efficiently at wavelengths above 4x or so the longest dimension of the opening.
Also, although each opening might not radiate efficiently, an array of holes in the case can produce constructive interference, leading to substantial radiation at certain frequencies, which may be what your friend was thinking of when he mentioned the number of holes affecting the radiation.
If all of the above are the case, is there perhaps some lining I could apply to non-metallic enclosures to control stray EMI?
Yes. You could either build a shielding enclosure around your circuit and then install the whole shield into the externally-visible case, or you could apply a metallic coating to the inside of a plastic or other non-metallic case. For example, Laird Technologies offers Eccoshield ES conductive lacquer coating. I've never priced these materials, but I suspect that they will cost more than simply using a metal case.
One trick with a paint-on coating is that typically you want the shield to be connected to the circuit ground or an external ground by some means, so you'll need to have some kind of reliable contact to the shield.