Power Spectral Density as a single number confusion
Question 1: Yes you are correct, the power spectral density is the power distribution per unit frequency so is a continuous function of frequency.
Question 2: The single number as given can be an estimate of total power.
What they gave is completely incorrect starting with the formula as given:
$$P = lim_{T \rightarrow \infty}\frac{1}{T}\int_0^T|x(k)|^2dt$$
This is an attempt to provide a formula for the total power in the signal, as the time average of the energy of the signal (but where is t in the function for x?). I assume $k$ is a typo, but if it refers to an FFT bin, then as $T \rightarrow \infty$ discrete $x(k)$ would become a continuous function of time as $x(t)$ so that the above would properly represent the total power as:
$$P = lim_{T \rightarrow \infty}\frac{1}{T}\int_0^T|x(t)|^2dt$$
Note that the absolute value squared of the FFT bins, $|x(k)|^2$ can be used as an estimate of the power spectral density (see links below for conditions on this), with proper scaling to any units desired as the the relative power in each bin.
For further details on estimating PSD from the DFT, see:
Power spectral density vs. FFT bin magnitude
What is the difference between PSD and squared magnitude of frequency spectrum?