every explanation of birefringence I find talks about unpolarized light entering Calcite, but what if I use a polarized light? Will that mean that as I rotate the crystal the light will disappear then reappear, etc?
If you put unpolarized light in your Calcite crystal, then the beam will be divided into two beams, one for each polarization, with equal intensity.
It you put polarized light (say, linearly), then the two beams will have an intensity that depends on the angle between the axis of polarization of the light and the crystal, following Malus' law. If the entering light is polarized along one of the axis of the crystal, then only one beam will emerge.
To answer your last question, if you rotate the crystal, then there will be two beams of varying intensity, the sum of the intensities being constantly equal to the incoming intensity. Periodically one of the beams disappears.