This sketch is only a PFD type arrangement. The full details would be seen on the PID.
a)does the liquid going to the reflux drum remain fully in the saturated liquid phase?
Often, the liquid going to the reflux drum is slightly subcooled for a total condensor application. This is the only way one can be sure the overhead vapor stream is completely condensed. Some portion of the reflux condensor HX tubes are fully submerged in the condensed liquid to enable this subcooling and process controls will be in place for this
b) Is any vapor formed in the reflux drum?
No additional vapor is formed. But there is often a hot vapor vapor bypass around the reflux condensor leading into the reflux drum to enable pressure control. This is necessary to prevent the reflux drum pressure from going too low since liquid here is partly subcooled. See PID for full details - often a dPIC controller performs this function
c)Is the reflux drum usually just an accumulator or does it also act as a separator?
Since vapor flow into the drum has only a small flow of hot bypass vapor going to it, it acts more as an accumulator than a separator
d)If the reflux drum has a boot, does this mean there's another liquid phase, making it a three-phase separator?
That is usually the case. To be sure, see the PID.