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I am following Baumann's notes on Cosmology (available here) and on page 78 the author defines the concept of adiabatic perturbations as having pressure and density fluctuations connected by $\delta P = c_s^2 \delta \rho$, where $c_s$ is identified as the speed of sound. What is the justification for naming $c_s$ in this way?

Later, on page 81, it is stated that $c_s = 0$ for linearised CDM (cold dark matter) fluctuations. Can someone provide a mathematical justification for this result?

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That is the standard definition of the speed of sound as long as you are dealing with adiabatic perturbations. You might want to check this. On the other hand, for CDM perturbations $c_s=0$ because CDM is collisionless and therefore, pressureless which makes $c_s=0$

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