The electric and magnetic fields are always in-phase if the wave can be treated as a plane wave (which simply means it cannot be too close to the source), and in vacuum or any medium with linear response, such as air.
Boundary conditions of wave guides change this relationship, and must be solved for each specific case. If the wave guide is large enough, you will only see effects near the surfaces.
Transmission through a conductor results in phase-lag of the magnetic field, and a rapid extinction of the propagating wave; see skin depth.
The methods used are nicely described here: Chapter 9: Electromagnetic Waves - MIT OpenCourseWare
Electromagnetic plane wave: electric and magnetic fields are always in phase.
You can study the mathematical proof, based on Maxwell's equations for an arbitrary plane wave, here.
So the direct answers are:
(A) No, except that they are always in phase in the "far field";
(B) No, see (A).
In the near field, for a dipole antenna, see Chapter 10: Antennas and Radiation - MIT OpenCourseWare, especially section 10.2.
This is why you ordinarily only need to consider the electric field with radio or light transmission, except in a wave guide, or non-linear media.
Occasionally somebody claims that the electric and magnetic fields are out of phase for circular polarization. This is not quite correct: the quarter wave plate accepts linearly polarized light at some angle wrt the fast axis of the QWP; the orthogonal components of the electric field vector under go different amounts of optical delay, resulting in two distinct electromagnetic waves, with the same phase delay in their electric and magnetic fields. See this animation of circular polarization; the input light is linearly polarized, with the red and blue representing the two components wrt the fast axis of the quarter wave plate, QWP.