We can have real photons as well as virtual photons. However, I think we can have only virtual weak gauge bosons $W^{\pm},Z$. Is that correct?

If virtual particles are just a manner of interpreting the S-matrix elements at various orders, why something as physical as energy-time uncertainty principle is invoked to explain their fleeting existence? The mathematics of quantum field theory, as far as I know, doesn't give any such picture to take this description seriously.

Since, the virtual particles do not obey dispersion relations, how justified is the measurement of Z-boson mass using the four-momentum conservation as explained in the maximum voted answer here?