Is it valid to calculate the magnitude, power, and phase of a real time-domain signal without converting it to the frequency-domain? I would assume you perform the calculations the same way, but since there would be no imaginary component because the signal is not complex it would be simpler:

magnitude = |x(n)|

power = |x(n)|^2

phase = 0 (for all values of n) - since there's no imaginary portion the phase equation would always result in a 0 value.

I'm trying to fully grasp and internalize some of the basic concepts of signal processing, and while I haven't ever really seen anyone try to use the equation for phase on a time-domain signal I don't see why it wouldn't be valid even if the results don't have much meaning.
No, this doesn't make much sense. What you can do in the time domain is compute the analytic signal and derive the signal's instantaneous amplitude (envelope) and its instantaneous phase from it.

Take as an example

$$x[n]=A\sin(\omega_0n+\theta)\tag{1}$$

The corresponding analytic signal is

$$x_a[n]=-jAe^{j(\omega_0n+\theta)}\tag{1}$$

Its instantaneous amplitude (envelope) is $\big|x_a[n]\big|=A$, and its instantaneous phase is $\arg\{x_a[n]\}=\omega_0n+\theta-\pi/2$.