Does the A train stop at the Port Authority Bus Terminal?

Yes, the A train stops at the Port Authority Bus Terminal. All three of the trains on the blue line (the A, the C and the E) stop at 42nd Street-Port Authority Bus Terminal.

There is also an underground tunnel connecting the 42nd Street-Port Authority stop on the A-C-E line to the 42nd Street-Times Square Station (since the A-C-E runs up and down 8th Avenue, while the Times Square Station is at 7th Avenue), so you can walk underground from the 42nd Street-Port Authority Station to the 42nd Street-Times Square Station, without exiting the subway system.