Question on understanding the time-bandwidth of a signal
This is not a rigorous answer.

First, why is the time-bandwidth product equal to the number of
degrees of freedom and what is the number of degrees of freedom of a
signal? Does it hold for both the deterministic and random signal?

The degree of freedom is the number of samples needed to represent a complex continuous-time signal. The Nyquist Shannon sampling theorem states that for any signal confined within $\left[-\frac{\Delta f}{2},+\frac{\Delta f}{2}\right]$, we need at least $\Delta f$ sample per time unit to perfectly reconstruct it.
If in the time domain, the signal can be approximated by its duration $\left[-\frac{\Delta t}{2},+\frac{\Delta t}{2}\right]$, the number of samples needed is thus $\Delta f \Delta t$, which is the time-bandwidth product defined in [Ozaktas et al.].

Second, why is there a (minimum) number of complex numbers required to
characterize a signal? What if the number is not reached?

If the number is not reached, the discrete-time signal represented by the insufficient number of samples cannot be used to reconstruct the continuous-time signal. An aliasing will be observed, for example.