For the stationary case with time varying magnetic field Maxwell's equations produce an induced electrical field which will accelerate charges. For the case of moving conductor and stationary field, a Lorentz transform of the constant magnetic field into the frame of the moving conductor will, again, produce an induced electrical field. In both cases the forces on charges are caused by the electric field component.
A constant magnetic field in the coordinate system of moving charges will therefor always produce an effective electric field component, which will result in an acceleration of charged particles. This can be shown very nicely in experiments, in which electron beams get deflected by a constant magnetic field.
There is no way to tell, for there is not such thing as a perfect conductor. Presumably, the magnetic field would be frozen at the configuration existing when the conductor was first created. In actual conductors, however, the conductivity value may be large, but always finite, and the the magnetic fields inside them diffuse at timescales proportional to this value.
For simplicity, let us assume a conductor with uniform conductivity and light propagation speed.
From charge conservation, we get $$\frac{\partial\rho }{\partial t} = - \nabla\cdot\mathbf{J}.$$
Since $\mathbf{J}=\sigma\mathbf{E}$, and $\nabla\cdot\mathbf E=4\pi\rho$, this leads to
$$\frac{\partial \rho}{\partial t}=-4\pi\sigma\rho,$$
so that the charge density relaxes at timescales inversely proportional to the conductivity (fast). As I've stated, but not yet proved, the magnetic field relaxes at much slower timescales, so it is safe to assume that the charge density is always equal to zero.
From Faraday's law
$$
\nabla \times \mathbf E =- \frac{\partial B}{\partial t}
$$
We see that the slowly varying magnetic field induces a slowly varying solenoidal electric field, so that $$\left|\frac{\partial \mathbf E}{\partial t}\right|\ll \sigma \mathbf E$$. Therefore, Ampere's law gives us
$$\nabla \times \mathbf B = \frac{1}{c}\left[4\pi\mathbf J+\frac{\partial \mathbf E}{\partial t}\right] = \frac{1}{c}\left[4\pi \sigma \mathbf E+\frac{\partial \mathbf E}{\partial t}\right] \approx \frac{4\pi \sigma }{c}\mathbf E, $$
or $$\nabla\times\nabla\times\mathbf{B}= -\frac{4\pi \sigma }{c} \frac{\partial \mathbf B}{\partial t}. $$
From the vector identity $$\nabla\times\nabla\times\mathbf{B} = \nabla(\nabla \cdot \mathbf{B}) - \nabla^2 \mathbf B $$ and Gauss's law for the magnetic field $\nabla \cdot \mathbf B = 0$, we get
$$
\nabla^2 \mathbf B = \frac{4\pi \sigma }{c} \frac{\partial \mathbf B}{\partial t},
$$
which is a diffusion equation for the magnetic field. In an initial configuration with spatial variation scale $L$ the characteristic relaxation time is $$\tau_D \sim \frac{4\pi \sigma }{c L^2}.$$ In the limit of a perfect conductor, $\tau_D$ would go to infinity. In realistic cases, it varies vastly. I quote from Jackson's "Classical Electrodynamics":
For a copper sphere of radius 1cm, the decay time of some initial $\mathbf B$ field inside is of the order of 5-10 miliseconds; for the molten iron core of the earth it is of the order of $10^5$ years.
When the medium is a fluid, or a plasma, the magnetic field undergoes induction as well as diffusion. The relative effects of the two processes are often characterized by the magnetic Reynolds number.
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Edit: I realized I misunderstood the question - I'm used to dealing with superconducting coils, so my answer referred to the flux through the middle of a coil of superconductor.
No - whatever flux existed through the middle of a coil of superconductor when it became a superconductor (usually, when it was cooled below its $T_c$) is locked into the coil. Any additional magnetic flux changes around the superconductor are expelled by inducted EMFs.
In superconducting magnets, usually the superconducting coils are charged by induction while they are above $T_c$ (and therefore not yet superconducting), then they are cooled down; as long as they stay cool and in the superconducting state, the flux through the coil will not change, and the coil acts like a permanent magnet.
Also, it sounds like you think protons are moving around in superconductors, which is not the case - just the electrons.