How electric fields and magnetic fields are connected? I'm not a physicist, but I have been reading several sources of how electric fields and magnetic fields are connected.
One question in particular (How do moving charges produce magnetic fields?) caught my attention
and have two well written answers:
The accepted answer of this question concludes:

We can conclude that magnetism is nothing more than electrostatics
combined with special relativity.

So I have two observations:

*

*The drift speed of a electron through a copper wire of cross-sectional area 3.00 x 10-6 m2, with a current of
10 A will be approximately 2.5 x 10-4 m/s, a very low relativistic speed.
The velocity of electrons in a wire isn't very slow to show these effects?


*You can also produce a electric field moving a magnet Does a moving magnetic field produce an electric field?. You can even produce a electromagnetic wave if you accelerate a magnet [Is it possible to create magnetic waves?], the same behavior that a charged particle have. Doesn't this show that electromagnetic field have two real components?
So, my question is: don't these questions are better answered with quantum mechanics and its theory of fields [like this one]?
 A: I would go even further and claim: 
Magnetism is just a relativistic effect.
Let's address your remarks:

The velocity of electrons in a wire isn't very slow to show these effects?

"Relativistic" does not necessarily mean "moving close to speed of light". It just means that you have to take into account relativity, i.e. how physics works in different frames.  For example:

You can also produce a electric field moving a magnet [...]. You can even produce a
electromagnetic wave if you accelerate a magnet [...]

Well, but if you were moving in a frame of reference at the same speed of the magnet (first part of the sentence) or accelerating at the same rate of the magnet (second part), so that the magnet always looked stationary, would you still see the emission of EM waves?
Also:

Aren't these questions better answered with quantum mechanics and
its theory of fields [like this one]?

The top answer to that question you link actually doesn't use quantum mechanics. Not even quantum field theory. Hence you can still claim it's a fully classical treatment.
Same physics, different frames
You will have already seen the derivation for the magnetic field of a wire from special relativity (here). Basically in a frame that is stationary with respect to the wire, you have zero net electric charge, so zero electric field $\mathbf{E}$ and hence a Lorentz force of $\mathbf{F} = q\mathbf{E} = \mathbf{0}$.
However, in a frame of reference moving at speed $\mathbf{v}$ in the direction of current (and this is where the relativistic comes in), length contraction results in a net charge density and hence a non-zero electric field! So in that frame, $\mathbf{F}' = q\mathbf{E}' \neq \mathbf{0}$.
But physics should be the same in all frames. So you must add something to the Lorentz force in your stationary force, which is just the Lorentz-transformed $\mathbf{E}'$ which you call $\mathbf{v}\times \mathbf{B}$. The $\mathbf{v}$ comes from the change of frame.
Again, no quantum effects here.
Quantum?
The issue with the previous statement is the following. Not all magnetic fields "go away" with a certain frame change.
For example - what about magnets??
Indeed, the intrinsic magnetic properties of a material are due, microscopically, to spin. This requires a quantum treatment. Or even better, a quantum field theoretical treatment.
But my statement Magnetic is just a relativistic effect still holds, since spin "comes out" of the (relativistic) Dirac equation.
