Dr Paul Leyland

a) I would recommend NOT using median stacking.  Averaging and summing are essentially the same operation, just that the first divides by the number of images stacked.  Median stacking is not a linear operation and (almost?) all photometry assumes a linear detector response.

b) The exposure of the stacked frames is the sum of the exposures of the individual frames.

c) I believe you should use the weighted mean of the mid-point times of the individual frames, where the weights are the exposure times of each frame.  If your stack consists of equally spaced exposures of equal exposure times, the mid-point of the central frame (assuming it exists, what if you have an even number of frames?) is the same thing.

Something to be aware of if you want the highest accuracy: all the above assumes that the focus and the sky background do not vary greatly throughout the stack.  If you are in doubt, perform the photometry on each frame to give a flux measurement (not the magnitude, which varies as the logarithm of the flux) and its corresponding variance.  Then add up all the fluxes weighted by their corresponding variance .  The stacked variance is  the root-mean-square of the individual values.

All the above assumes you are doing differential photometry — that you are measuring the relative brightness of a target and a supposedly constant comparison star.