There is a very slight difference in the dielectric constants of space and air. The dielectric constant of space is one. The dielectric constant of air at sea level is 1.000536. In all but the highest precision calculations, the slight difference is neglected. That is, it radiates uniformly in all directions.
P t is either peak or average power depending on how P D is to be specified.
Radars use directional antennas to channel most of the radiated power in a particular direction. Power density from radar, P t is either peak or average power depending on how P D is to be specified. If the power density at a specified range is one microwatt per square meter and the antenna's effective capture area is one square meter then the power captured by the antenna is one microwatt. That's nothing more than converting the power and range to the proper units. However, antenna gain is almost always given in dB, not as a ratio.