I was taking a look through Lecture 13 and noticed that for the density of states (and everything used to calculate it in that lecture), there is no factor L^3. Is this a mistake, is everything defined per unit of volume instead or is there another reason?
This isn’t a mistake, and indeed we consider everything per unit volume.
Is there a specific one you need to use for the exam, or is either fine as long as you specify which (provided it is not specified on the exam, like in 2019 3a).
Either is fine as long as you are consistent.