Allow me to take a stab at this and try not to introduce too many new issues.
1) The 'phase' can be viewed as the 'delay' between the input signal and the output signal. When considering scattering parameters, the signal being measured is usually a 'voltage'. Any 'delay' indicates the presence of reactance in the circuit. (OK, I have to complicate this a bit. For one port devices where there is only one electrical node one can measure, one can still think of an 'input signal' and an 'output signal' which are simply 'added' to produce the measured signal.)
2) The terms 'real part' and 'imaginary part' are meaningless if no 'of' is included. For example, I may report a measurement has a "real part of .1". The question is, are we talking about the 'real part of the reflection coefficient' or the 'real part of the impedance'. Stated simply, "No 'of', no meaning".
As for the Smith chart, the plot is one of the 'real/imaginary parts of the voltage reflection coefficient' but the circles are the 'real and imaginary parts' of the impedance. Usually, a marker reports the 'real/imag' part of the impedance but that is wholly product dependent.
3) They MAY be the same, it depends on the 'of'...... If the real/imag refer to the voltage reflection coefficient then they will be different than the impedance. But... if real/imag is 'of' the impedance, they would be the same (by definition).
4) Spp parameters are ratios and are therefore unitless. The vast majority of Scattering Parameter papers discuss the ratios of VOLTAGES. A few use the term to refer to Power Ratios. ( I implore people to avoid the trap of arguing the 'proper' definition. I understand there are two camps. I have found it to be a fruitless discussion as it generally devolves into a 'Holy War'.) For the newly initiated, one can safely assume Spp refers to a voltage ratio.
Hope that helps.
