Is it valid to calculate the magnitude, power, and phase of a real time-domain signal without converting it to the frequency-domain?
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 $\arg\{x_a[n]\}=\omega_0n+\theta-\pi/2$.