Transistors' Scattering Parameters
answers


Yes.   
Because of the assumption to make v1=v2=v3.    


But  generally we do not test in this condition. 
We only apply 1 voltage or current  to “a” to get the response “b” on all ports 
such that ... \$S_{xy}=\dfrac{b_x}{a_y}\$ for each frequency tested or in a continuous display

That’s a math problem others can explain better. I think this is to prove conservation of energy and any ratio lost or gained adds up a net power of 0 and a net forward/reverse ratio sum of 1 means there is no net signal lost/generated/reflected as a test to the equation with 0 inputs. 


Other:

Three-port S3P files need 19 columns of magnitude and angle data. The data in the Scattering Matrix vectors must be wrapped onto three lines:

For EDA software    Insert comments    To ensure correct order.
    ! Freq S11M S11A S21M S21A S12M S12A S22M S22A

S11m S11a S12m S12a S13m S13a
S21m S21a S22m S22a S23m S23a
S31m S31a S32m S32a S33m S33a